Normal +Bionomial

882019 Normal +Bionomial

httpslidepdfcomreaderfullnormal-bionomial 180

Probability Distribution

Random variables

Discrete

Continuous

Applications

Binomial Tables Applications



AA random variablerandom variable is a numerical description of theis a numerical description of theoutcome of an experimentoutcome of an experimentAA

random variablerandom variable is a numerical description of theis a numerical description of theoutcome of an experimentoutcome of an experiment

Random VariablesRandom Variables

AA discrete random variablediscrete random variable may assume either amay assume either a

finite number of values or an infinite sequence of finite number of values or an infinite sequence of valuesvalues



AA continuous random variablecontinuous random variable may assume anymay assume any

numerical value in an interval or collection of numerical value in an interval or collection of intervalsintervals





LetLet x x = number of TVs sold at the store in one day= number of TVs sold at the store in one day

wherewhere x x can take on 5 values (0 1 2 3 4)can take on 5 values (0 1 2 3 4)

Le

tLet x x = number of TVs sold at the store in one day= number of TVs sold at the store in one day


Example JSL Appliances

Discrete random variable with a finite number of values



LetLet x x = number of customers arriving in one day= number of customers arriving in one day

wherewhere x x can take on the values 0 1 2 can take on the values 0 1 2



Example JSL AppliancesExample JSL Appliances

s Discrete random variable with anDiscrete random variable with an infiniteinfinite

sequence of valuessequence of values

We can count the customers arriving but there is noWe can count the customers arriving but there is no

finite upper limit on the number that might arrivefinite upper limit on the number that might arrive




QuestionQuestion Random VariableRandom Variable x x TypeType

FamilyFamily

sizesize

x x = Number of dependents= Number of dependents

reported on tax returnreported on tax returnDiscreteDiscrete

Distance fromDistance fromhome to storehome to store

x x = Distance in miles from= Distance in miles fromhome to the store sitehome to the store site

ContinuousContinuous

Sell anSell an

automobileautomobile Gender of the customer Gender of the customer

x x = 0 if male= 0 if male= 1 if female= 1 if female

DiscreteDiscrete



The The probability distribution

probability distribution for a random variablefor a random variabledescribes how probabilities are distributed overdescribes how probabilities are distributed over

the values of the random variablethe values of the random variable




We can describe a discrete probability distributionWe can describe a discrete probability distributionwith a table graph or equationwith a table graph or equation


Discrete Probability DistributionsDiscrete Probability Distributions



The probability distribution is defined by a The probability distribution is defined by aprobability functionprobability function denoted by denoted by f f (( x x ) which provides) which provides

the probability for each value of the random variablethe probability for each value of the random variable



The required conditions for a discrete probability The required conditions for a discrete probabilityfunction arefunction are



f f (( x x )) gtgt 00f f (( x x )) gtgt 00

ΣΣ f f (( x x ) = 1) = 1ΣΣ f f (( x x ) = 1) = 1



s a tabular representation of the probabilitya tabular representation of the probabilitydistribution for TV sales was developeddistribution for TV sales was developed

s Using past data on TV sales hellipUsing past data on TV sales hellip

NumberNumber

Units SoldUnits Sold of Daysof Days00 8080

11 5050

22 4040

33 101044 2020

200200

x x f f (( x x ))00 4040

11 2525

22 2020

33 050544 1010

100100

8020080200




1010

2020

3030

4040

5050

0 1 2 3 40 1 2 3 4

Values of Random VariableValues of Random Variable x x (TV sales)(TV sales)Values of Random VariableValues of Random Variable x x (TV sales)(TV sales)

P r o

b a

b i l i t

y

P r o

b a

b i l i t

y

P r o

b a

b i l i t

y

P r o

b a

b i l i t

y


Graphical Representation of Probability Distribution



xpec e a ue anVariance

The The expected valueexpected value or mean of a random variable or mean of a random variableis a measure of its central locationis a measure of its central locationThe The expected valueexpected value or mean of a random variable or mean of a random variableis a measure of its central locationis a measure of its central location

The The variancevariance summarizes the variability in thesummarizes the variability in the

values of a random variablevalues of a random variable



The The standard deviationstandard deviation σ σ is defined as the positive is defined as the positive

square root of the variancesquare root of the variance



Var(Var( x x ) =) = σ σ 22 == ΣΣ (( x x -- micro micro ))22f f (( x x ))Var(Var( x x ) =) = σ σ 22 == ΣΣ (( x x -- micro micro ))22f f (( x x ))

EE(( x x ) =) = micro micro == ΣΣ xf xf (( x x ))EE(( x x ) =) = micro micro == ΣΣ xf xf (( x x ))



Expected Value

expected numberexpected number

of TVs sold in a dayof TVs sold in a dayexpected numberexpected number

of TVs sold in a dayof TVs sold in a day

x x f f (( x x )) xf xf (( x x ))

00 4040 0000

11 2525 2525

22 2020 4040

33 0505 1515

44 1010 4040

EE(( x x ) = 120) = 120




Variance and Standard Deviation

00

1122

33

44

-12-12

-02-020808

1818

2828

144144

004004064064

324324

784784

4040

2525

2020

0505

1010

576576

010010

128128

162162

784784

x - x - micro micro (( x - x - micro micro ))22 f f (( x x )) (( x x -- micro micro ))22f f (( x x ))

Variance of daily sales =Variance of daily sales = σ σ 22 = 1660= 1660

x x

TVs TVs

squaresquare

dd

Standard deviation of daily sales = 12884 TVsStandard deviation of daily sales = 12884 TVs




Continuous Probability Distributions

Uniform Probability Distribution

Normal Probability Distribution Normal Approximation of Binomial Probabilities

Exponential Probability Distribution

f ( x )f ( x )

x x

UniformUniform

x x

f f (( x x ))NormalNormal

x x

f ( x )f ( x ) ExponentialExponential




s AA continuous random variablecontinuous random variable can assume anycan assume any

value in an interval on the real line or in avalue in an interval on the real line or in acollection of intervalscollection of intervals

s It is not possible to talk about the probabilityIt is not possible to talk about the probability

of the random variable assuming a particularof the random variable assuming a particular

valuevalues Instead we talk about the probability of theInstead we talk about the probability of the

random variable assuming a value within arandom variable assuming a value within a

given intervalgiven interval



Continuous Probability DistributionsContinuous Probability Distributions

s The probability of the random variable The probability of the random variable

assuming a value within some given intervalassuming a value within some given intervalfromfrom x x 11 toto x x 22 is defined to be theis defined to be the area underarea under

the graphthe graph of theof the probability density functionprobability density function betweenbetween x x

11 andand x x 22

f ( x )f ( x )

x x

UniformUniform

x x 11 x x 11 x x 22 x x 22

x x


x x 11 x x 11 x x 22 x x 22

x x 11 x x 11 x x 22 x x 22

ExponentialExponential

x x

f ( x )f ( x )

x x 11

x x 11

x x 22 x x 22



n orm ro a yDistribution

wherewhere aa = smallest value the variable can assume= smallest value the variable can assume

bb = largest value the variable can assume= largest value the variable can assume

f f (( x x ) = 1() = 1(bb ndashndash aa) for) for aa ltlt x x ltlt bb

= 0 elsewhere= 0 elsewhere



s A random variable isA random variable is uniformly distributeduniformly distributed

whenever the probability is proportional to thewhenever the probability is proportional to theintervalrsquos lengthintervalrsquos length

s The The uniform probability density functionuniform probability density function isis



Var(Var( x x ) = () = (bb -- aa))221212Var(Var( x x ) = () = (bb -- aa))221212

E(E( x x ) = () = (aa ++ bb)2)2E(E( x x ) = () = (aa ++ bb)2)2

n orm ro a yDistributions Expected Value of Expected Value of x x

s Variance of Variance of x x



n orm ro a yDistributions Example Slaters BuffetExample Slaters Buffet

Slater customers are charged for theSlater customers are charged for theamount of amount of

salad they take Sampling suggests that thesalad they take Sampling suggests that theamountamount

of salad taken is uniformly distributedof salad taken is uniformly distributedbetween 5between 5

ounces and 15 ouncesounces and 15 ounces



s Uniform Probability Density FunctionUniform Probability Density Function

f f (( x x ) = 110 for 5) = 110 for 5 ltlt x x ltlt 1515




wherewhere

x x = salad plate filling weight= salad plate filling weight

Uniform Probability DistributionUniform Probability Distribution



s Expected Value of Expected Value of x x


E(E( x x ) = () = (aa ++ bb)2)2

= (5 + 15)2= (5 + 15)2

= 10= 10

E(E( x x ) = () = (aa ++ bb)2)2

= (5 + 15)2= (5 + 15)2

= 10= 10

Var(Var( x x ) = () = (bb -- aa))221212

= (15 ndash 5)= (15 ndash 5)22

1212= 833= 833

Var(Var( x x ) = () = (bb -- aa))221212

= (15 ndash 5)= (15 ndash 5)22

1212= 833= 833


n orm ro a




for Salad Plate Filling Weight

f ( x )f ( x )

x x 55 1010 1515

110110

Salad Weight (oz)Salad Weight (oz)




f ( x )f ( x )

x x 55 1010 1515

110110


P(12 lt x lt 15) = (110)(3) = 3P(12 lt x lt 15) = (110)(3) = 3

What is the probability that a customerWhat is the probability that a customer

will take between 12 and 15 ounces of will take between 12 and 15 ounces of

saladsalad

1212




Two major difference between the continuous random

variables and their discrete counterparts

bull We can take out probability of the random variable

assuming a value within a given interval but probability of

the random variable assuming a particular value cannot findout

bull Probability of a continuous random variable assuming any

particular value exactly is zero Because the area under the

graph of f(x) at a single point is zero

orma ro a y



orma ro a yDistribution The normal probability distribution is the most important distribution for

describing a continuous random variable It is widely used in statistical inference

It is also known as Gaussion distribution

s It has been used in a wide variety of It has been used in a wide variety of

applicationsapplicationsbull Heights of peopleHeights of people

bull Scientific measurementsScientific measurements

bull Test scores Test scores

bull Amounts of rainfallAmounts of rainfall

orma ro a y



orma ro a yDistributionNormal Probability Density Function

2 2( ) 21( )

2

x f x e micro σ

σ π

minus minus

=

micro micro = mean= mean

σ σ = standard deviation= standard deviation

π π = 314159= 314159

ee = 271828= 271828

wherewhere



The distribution is The distribution is symmetricsymmetric its skewness its skewness

measure is zeromeasure is zero The distribution is The distribution is symmetricsymmetric its skewness its skewness

measure is zeromeasure is zero

Normal Probability DistributionNormal Probability Distribution

s CharacteristicsCharacteristics

x x



The entire family of normal probability The entire family of normal probability

distributions is defined by itsdistributions is defined by its meanmean micro micro and itsand its

standard deviationstandard deviation σ σ






Standard DeviationStandard Deviation σ σ

MeanMean micro micro x x



The The highest pointhighest point on the normal curve is at theon the normal curve is at the

meanmean which is also the which is also the medianmedian andand modemode The The highest pointhighest point on the normal curve is at theon the normal curve is at the

meanmean which is also the which is also the medianmedian andand modemode



x x





-10-10 00 2020

The mean can be any numerical value negative The mean can be any numerical value negative

zero or positivezero or positive



x x





σ = 15= 15

σ = 25= 25

The standard deviation determines the width of the The standard deviation determines the width of the

curve larger values result in wider flatter curvescurve larger values result in wider flatter curves The standard deviation determines the width of the The standard deviation determines the width of the

curve larger values result in wider flatter curvescurve larger values result in wider flatter curves

x x



Probabilities for the normal random variable areProbabilities for the normal random variable are

given bygiven by areas under the curveareas under the curve The total area The total area

under the curve is 1 (5 to the left of the mean andunder the curve is 1 (5 to the left of the mean and

5 to the right)5 to the right)







55 55

x x





of values of a normal random variableof values of a normal random variable

are within of its meanare within of its mean



6826682668266826

+- 1 standard deviation+- 1 standard deviation+- 1 standard deviation+- 1 standard deviation

of values of a normal random variableof values of a normal random variableare within of its meanare within of its mean


9544954495449544+- 2 standard deviations+- 2 standard deviations+- 2 standard deviations+- 2 standard deviations





9972997299729972

+- 3 standard deviations+- 3 standard deviations+- 3 standard deviations+- 3 standard deviations





x x micro micro ndashndash 33σ σ micro micro ndashndash 11σ σ

micro micro ndashndash 22σ σ

micro micro + 1+ 1σ σ


micro micro + 3+ 3σ σ micro micro

68266826

95449544

99729972

Standard Normal Probability



Standard Normal ProbabilityDistribution

A random variable having a normal distributionA random variable having a normal distributionwith a mean of 0 and a standard deviation of 1 iswith a mean of 0 and a standard deviation of 1 is

said to have asaid to have a standard normal probabilitystandard normal probability

distributiondistribution






σ σ = 1= 1

00

z z

The letter The letter z z is used to designate the standardis used to designate the standardnormal random variablenormal random variable The letter The letter z z is used to designate the standardis used to designate the standardnormal random variablenormal random variable

Standard Normal Probability DistributionStandard Normal Probability Distribution



s Converting to the Standard NormalConverting to the Standard Normal

DistributionDistribution


z x

=

minus micro

σ

We can think of We can think of z z as a measure of the number of as a measure of the number of standard deviationsstandard deviations x x is fromis from micro micro

Standard Normal Density FunctionStandard Normal Density Function





s Example Pep ZoneExample Pep Zone

Pep Zone sells auto parts and suppliesPep Zone sells auto parts and supplies

includingincluding

a popular multi-grade motor oil When thea popular multi-grade motor oil When the

stock of stock of

this oil drops to 20 gallons a replenishmentthis oil drops to 20 gallons a replenishment

order isorder is

placedplaced



The store manager is concerned that sales The store manager is concerned that sales

are beingare being

lost due to stockouts while waiting for an orderlost due to stockouts while waiting for an order

It haIt ha

been determined that demand duringbeen determined that demand during

replenishmentreplenishment

lead-time is normally distributed with a mean of lead-time is normally distributed with a mean of

1515

gallons and a standard deviation of 6 gallonsgallons and a standard deviation of 6 gallons

The manager would like to know the The manager would like to know the

probability of probability of

a stockouta stockout PP(( x x gt 20)gt 20)





z z = (= ( x x -- micro micro ))σ σ

= (20 - 15)6= (20 - 15)6= 83= 83


= (20 - 15)6= (20 - 15)6= 83= 83

s Solving for the Stockout ProbabilitySolving for the Stockout Probability

Step 1 ConvertStep 1 Convert x x to the standard normal distributionto the standard normal distributionStep 1 ConvertStep 1 Convert x x to the standard normal distributionto the standard normal distribution

Step 2 Find the area under the standard normalStep 2 Find the area under the standard normal

curve to the left of curve to the left of z z = 83= 83



see next slidesee next slide see next slidesee next slide




PP(( z z gt 83) = 05 ndashgt 83) = 05 ndash PP(( z z ltlt 83)83)

= 05- 2967= 05- 2967

= 2033= 2033

PP(( z z gt 83) = 05 ndashgt 83) = 05 ndash PP(( z z ltlt 83)83)= 05- 2967= 05- 2967

= 2033= 2033


Step 3 Compute the area under the standard normalStep 3 Compute the area under the standard normal

curve to the right of curve to the right of z z = 83= 83



ProbabilityProbabilityof a stockoutof a stockout PP(( x x gt 20)gt 20)


d d l b bili i ib i



Standard Normal Probability Distribution

If the manager of Pep Zone wants the

probability of a stockout to be no more than

05 what should the reorder point be


S d d l b bili i ib i



s Solving for the Reorder PointSolving for the Reorder Point

00

Area = 9500Area = 9500

Area = 0500Area = 0500

z z z z

0505


d d l b bili i ib iS d d N l P b bili Di ib i




Step 2 ConvertStep 2 Convert z z 0505 to the corresponding value of to the corresponding value of x x Step 2 ConvertStep 2 Convert z z 0505 to the corresponding value of to the corresponding value of x x

x x == micro micro ++ z z 0505 σ σ

= 15 + 1645(6)= 15 + 1645(6)

= 2487 or 25= 2487 or 25


= 15 + 1645(6)= 15 + 1645(6)

= 2487 or 25= 2487 or 25

A reorder point of 25 gallons will place the probabilityA reorder point of 25 gallons will place the probabilityof a stockout during leadtime at (slightly less than) 05of a stockout during leadtime at (slightly less than) 05


S d d l b bili i ib iSt d d N l P b bilit Di t ib ti




By raising the reorder point from 20 gallons toBy raising the reorder point from 20 gallons to

25 gallons on hand the probability of a stockout25 gallons on hand the probability of a stockout

decreases from about 20 to 05decreases from about 20 to 05

This is a significant decrease in the chance that Pe This is a significant decrease in the chance that Pep

Zone will be out of stock and unable to meet aZone will be out of stock and unable to meet acustomerrsquos desire to make a purchasecustomerrsquos desire to make a purchase




Introduction and Concept

Based on Bernoulli Process

Is a discrete-time stochastic process consisting of a sequence of independent

random variables taking values over two symbols

We are not dealing with samples but with population values so dealing withparameters

Consider tossing a coin twice The possible outcomes are

no heads P (m = 0) = q2

one head P (m = 1) = qp + pq (toss 1 is a tail toss 2 is a head or toss 1 is head toss 2 is a tail) = 2 pq

two heads P(m = 2) = p2

Now recalling square of Binomial (p + q) is equal to the same as if added

above



Characteristics

Experiment consist of n identical trials

Each trial has only two outcomes

The probability of one outcome is p and the other is q=1-p

The probability stays the same from one trail to the next

The trials are statistically independent

We are interested in r the number of success observed during

the n trials



Binomial Formula

Binomial distribution the probability of r success out of N trials

( ) r N r r N r N

r

r N r

r N q pr N r

N q pq pC p N r P minusminusminus

minus===

)(

)(

Expectation Value micro = np = 50 13 = 16667

000

002

004

006

008

010

012

014

0 5 10 15 20 25 3k

P

( k

5 0

1 3 )

Expectation Value

micro = np = 7 13 = 2333

000

010

020

030

040

0 2 4 6 8 10k

P

( k

7

1 3 )

Bi i l Di t ib tiBi i l Di t ib ti



wherewhere f f (( x x ) = the probability of ) = the probability of x x successes insuccesses in nn trialstrials

nn = the number of trials= the number of trials

p p = the probability of success on any one trial= the probability of success on any one trial

( )( ) (1 )

( ) x n x n

f x p p x n x

minus= minusminus

Binomial DistributionBinomial Distribution

s Binomial Probability FunctionBinomial Probability Function



Binomial Formulas

Mean

Standard Deviation

( )

( ) ( )

( ) ( ) ( )

Np

p p N p

p pm p p p p N q pm p

pm N pq pm

q p p

N

m

m N m N m

N

m

m N m N m

N

m

m N m N m

N

m

m N m N m

N

m

m N m N m

N

m

m N m N m

=

minusminussdotminus=

sum minusminusminussum minusminus=sum

=sum minusminusminussum

=

sum

minusminusminus

=

minusminus

=

minusminus

=

minusminus

=

minusminus

=

minusminus

=

minus

micro

micro micro

part

part

111

0

1

0

1

0

1

0

1

0

1

0

)1(1)1(

)1()1()1()1(

0)1)((

0

σ 2=

(mminus micro )2P(m N p)m=0

N

sum

P(m N p)m=0

N

sum

=Npq

Bi i l Di t ib tiBinomial Distribution



( )( ) (1 )

( ) x n x n

f x p p x n x

minus= minusminus


s Binomial ProbabilityBinomial Probability

FunctionFunction

Probability of a particularProbability of a particular

sequence of trial outcomessequence of trial outcomes

withwith x x successes insuccesses in nn trialstrials




Number of experimentalNumber of experimental

outcomes providing exactlyoutcomes providing exactly

x x successes insuccesses in nn trialstrials








s Example Evans ElectronicsExample Evans Electronics

Evans is concerned about a low retentionEvans is concerned about a low retentionrate for employees In recent yearsrate for employees In recent years

management has seen a turnover of 10 of management has seen a turnover of 10 of

the hourly employees annually Thus for anythe hourly employees annually Thus for any

hourly employee chosen at randomhourly employee chosen at randommanagement estimates a probability of 01management estimates a probability of 01

that the person will not be with the companythat the person will not be with the company

next yearnext year





s Using the Binomial Probability FunctionUsing the Binomial Probability Function

Choosing 3 hourly employees at randomChoosing 3 hourly employees at randomwhat is the probability that 1 of them will leavewhat is the probability that 1 of them will leave

the company this yearthe company this year

f xn

x n x p p x n x( )

( )( )( )

=

minus

minusminus1

1 23(1) (01) (09) 3(1)(81) 2431(3 1)f = = =

minus

LetLet p p = 10= 10 nn = 3= 3 x x = 1= 1

Binomial Distribution



Tree Diagram


1st Worker1st Worker 2nd Worker2nd Worker 3rd Worker3rd Worker x x ProbProb

Leaves

(1)

Leaves

(1)

Stays(9)

Stays(9)

33

22

00

22

22

Leaves (1)Leaves (1)


S (9)S (9)

Stays (9)Stays (9)

Stays (9)Stays (9)

S (9)S (9)

S (9)S (9)

S (9)S (9)

L (1)L (1)

L (1)L (1)

L (1)L (1)

L (1)L (1) 00100010

00900090

00900090

72907290

00900090

11

11

08100810

08100810

08100810

1111




s Using Tables of Binomial ProbabilitiesUsing Tables of Binomial Probabilities

n x 05 10 15 20 25 30 35 40 45 50

3 0 8574 7290 6141 5120 4219 3430 2746 2160 1664 1250

1 1354 2430 3251 3840 4219 4410 4436 4320 4084 3750

2 0071 0270 0574 0960 1406 1890 2389 2880 3341 3750

3 0001 0010 0034 0080 0156 0270 0429 0640 0911 1250

p






(1 )np pσ = minus

EE(( x x ) =) = micro micro == npnpEE(( x x ) =) = micro micro == npnp

Var(Var( x x ) =) = σ σ 22

== npnp(1(1 minusminus p p))Var(Var( x x ) =) = σ σ 22

== npnp(1(1 minusminus p p))

s ExpectedExpected

ValueValue

s VariancVarianc

ee

s StandardStandard

DeviationDeviation





3(1)(9) 52 employeesσ = =

EE(( x x ) =) = micro micro = 3(1) = 3 employees out of 3= 3(1) = 3 employees out of 3EE(( x x ) =) = micro micro = 3(1) = 3 employees out of 3= 3(1) = 3 employees out of 3

Var(Var( x x ) =) = σ σ 22 = 3(1)(9) = 27= 3(1)(9) = 27Var(Var( x x ) =) = σ σ 22 = 3(1)(9) = 27= 3(1)(9) = 27

s Expected ValueExpected Value

s VarianceVariance

s Standard DeviationStandard Deviation



Applications

A committee consisting of 5 members votes on

whether or not to hire a new manager The

probability that each memberrsquos vote for

candidate A is 06 Only if over half of the

committee agrees to hire her does candidate Areceive the offer

1) What is the probability that the candidate A

gets an offer


does not get the offer



Answer

Selected = 03456+ 02592+ 00778

=06826

Not Selected = 1-06826 = 03174



Binomial Tables

Binomial Tables are used to same time

Binomial Tables Binom_Tabpdf



Applications


whether or not to hire a new professor The

probability that each memberrsquos vote for candidate

A is 06 Only if over half of the committee agrees

to hire her does candidate A receive the offer 1) What is the probability that the candidate A

gets an offer (Hint 06=04)



Poisson DistributionPoisson Distribution



A Poisson distributed random variable is oftenA Poisson distributed random variable is often

useful in estimating the number of occurrencesuseful in estimating the number of occurrences

over aover a specified interval of time or spacespecified interval of time or space




It is a discrete random variable that may assumeIt is a discrete random variable that may assumeanan infinite sequence of valuesinfinite sequence of values (x = 0 1 2 )(x = 0 1 2 )It is a discrete random variable that may assumeIt is a discrete random variable that may assume

anan infinite sequence of valuesinfinite sequence of values (x = 0 1 2 )(x = 0 1 2 )





Examples of a Poisson distributed random variableExamples of a Poisson distributed random variableExamples of a Poisson distributed random variableExamples of a Poisson distributed random variable

the number of knotholes in 14 linear feet of the number of knotholes in 14 linear feet of

pine boardpine board



the number of vehicles arriving at a tollthe number of vehicles arriving at a tollbooth in one hourbooth in one hour







s Two Properties of a Poisson Experiment Two Properties of a Poisson Experiment

22 The occurrence or nonoccurrence in any The occurrence or nonoccurrence in any

interval is independent of the occurrence orinterval is independent of the occurrence or

nonoccurrence in any other intervalnonoccurrence in any other interval


interval is independent of the occurrence orinterval is independent of the occurrence ornonoccurrence in any other intervalnonoccurrence in any other interval

11 The probability of an occurrence is the same The probability of an occurrence is the same

for any two intervals of equal lengthfor any two intervals of equal length



Poisson Distribution



Poisson Probability Function


f xe

x

x

( )

=

minus micro micro

wherewhere

f(x)f(x) = probability of = probability of x x occurrences in an intervaloccurrences in an interval

micro micro = mean number of occurrences in an interval= mean number of occurrences in an interval

ee = 271828= 271828





s Example Mercy HospitalExample Mercy Hospital

Patients arrive at the emergency room of Patients arrive at the emergency room of MercyMercy

Hospital at the average rate of 6 per hour onHospital at the average rate of 6 per hour on

weekendweekend

eveningsevenings

What is the probability of 4 arrivals in 30What is the probability of 4 arrivals in 30

minutes on a weekend eveningminutes on a weekend evening





s Using the Poisson Probability FunctionUsing the Poisson Probability Function

4 33 (271828)(4) 1680

4f

minus

= =

micro micro = 6hour = 3half-hour= 6hour = 3half-hour x x = 4= 4




s Poisson Distribution of ArrivalsPoisson Distribution of Arrivals


Poisson Probabilities

000

005

010

015

020

025

0 1 2 3 4 5 6 7 8 9 10

Number of Arrivals in 30 Minutes

P r o

b a

b i l i t

y

actuallyactuallythethesequencesequencecontinuescontinues

11 12 hellip11 12 hellip





A property of the Poisson distribution is thatA property of the Poisson distribution is thatthe mean and variance are equalthe mean and variance are equal


micro micro == σ σ 22





s Variance for Number of ArrivalsVariance for Number of Arrivals

During 30-Minute PeriodsDuring 30-Minute Periods

micro micro == σ σ 22 = 3= 3 micro micro == σ σ 22 = 3= 3



Example

Arrivals at a bus-stop follow a

Poisson distribution with an average

of 45 every quarter of an hour

Obtain a barplot of the distribution

(assume a maximum of 20 arrivals in

a quarter of an hour) and calculate

the probability of fewer than 3 arrivals

in a quarter of an hour



The probabilities of 0 up to 2 arrivals

can be calculated directly from theformula

( )

xe

p x x

λ λ minus

=

45 0

45(0)0

e pminus

=

with λ =45

So p(0) = 001111



Similarly p(1)=004999 and p(2)=011248

So the probability of fewer than 3 arrivals

is 001111+ 004999 + 011248 =017358

Normal ApproximationNormal Approximation



of Binomial Probabilitiesof Binomial Probabilities

When the number of trialsWhen the number of trials nn becomes large becomes large

evaluating the binomial probability function byevaluating the binomial probability function byhand or with a calculator is difficulthand or with a calculator is difficult

The normal probability distribution provides an The normal probability distribution provides an

easy-to-use approximation of binomialeasy-to-use approximation of binomial

probabilitiesprobabilitieswherewhere nn gt 20gt 20 npnp gtgt 5 and5 and nn(1 -(1 - p p)) gtgt 55





s SetSet micro micro == npnp

σ = minus(1 )np p

s Add and subtract 05 (aAdd and subtract 05 (a continuity correction factorcontinuity correction factor))

because a continuous distribution is being used tobecause a continuous distribution is being used to

approximate a discrete distribution For exampleapproximate a discrete distribution For example

PP(( x x = 10) is approximated by= 10) is approximated by PP(95(95 ltlt x x ltlt 105)105)

Exponential Probability DistributionExponential Probability Distribution



p yp y

s The exponential probability distribution is The exponential probability distribution is

useful in describing the time it takes touseful in describing the time it takes tocomplete a taskcomplete a task

s The exponential random variables can be used The exponential random variables can be used

to describeto describebull Time between vehicle arrivals at a toll booth Time between vehicle arrivals at a toll booth

bull Time required to complete a questionnaire Time required to complete a questionnaire

bull Distance between major defects in a highwayDistance between major defects in a highway




Density Function

p y

wherewhere micro micro = mean= mean ee = 271828= 271828

f x e x( ) =

minus1

micro micro

forfor x x gtgt 00 micro micro gt 0gt 0



Cumulative Probabilities


P x x ex

( )le = minus minus

0 1 o micro

wherewhere

x x 00 = some specific value of = some specific value of x x

Exponential ProbabilityDi t ib ti



Distributions Example Alrsquos Full-Service PumpExample Alrsquos Full-Service Pump

The time between arrivals of cars at Alrsquos The time between arrivals of cars at Alrsquosfull-servicefull-service

gas pump follows an exponential probabilitygas pump follows an exponential probability

distribution with a mean time between arrivalsdistribution with a mean time between arrivals

of of

3 minutes Al would like to know the probability3 minutes Al would like to know the probability

thatthat

the time between two successive arrivals will bethe time between two successive arrivals will be

22

minutes or lessminutes or less

xponent a ro a tyDi t ib ti



x x

f ( x )f ( x )

11

33

44

22

1 2 3 4 5 6 7 8 9 101 2 3 4 5 6 7 8 9 10 Time Between Successive Arrivals (mins) Time Between Successive Arrivals (mins)

Distribution

PP(( x x ltlt 2) = 1 - 2718282) = 1 - 271828-23-23 = 1 - 5134 = 486= 1 - 5134 = 486PP(( x x ltlt 2) = 1 - 2718282) = 1 - 271828-23-23 = 1 - 5134 = 486= 1 - 5134 = 486

Relationship between the Poissond E i l Di ib i



and Exponential Distributions

The Poisson distribution The Poisson distributionprovides an appropriate descriptionprovides an appropriate description

of the number of occurrencesof the number of occurrences

per intervalper interval




The exponential distribution The exponential distribution

provides an appropriate descriptionprovides an appropriate description

of the length of the intervalof the length of the interval

between occurrencesbetween occurrences







AA random variablerandom variable is a numerical description of theis a numerical description of theoutcome of an experimentoutcome of an experimentAA

random variablerandom variable is a numerical description of theis a numerical description of theoutcome of an experimentoutcome of an experiment














Le




















FamilyFamily

sizesize






Sell anSell an



DiscreteDiscrete






















ΣΣ f f (( x x ) = 1) = 1ΣΣ f f (( x x ) = 1) = 1





NumberNumber


11 5050

22 4040

33 101044 2020

200200

x x f f (( x x ))00 4040

11 2525

22 2020

33 050544 1010

100100

8020080200




1010

2020

3030

4040

5050

0 1 2 3 40 1 2 3 4


P r o

b a

b i l i t

y

P r o

b a

b i l i t

y

P r o

b a

b i l i t

y

P r o

b a

b i l i t

y



















Expected Value




x x f f (( x x )) xf xf (( x x ))

00 4040 0000

11 2525 2525

22 2020 4040

33 0505 1515

44 1010 4040

EE(( x x ) = 120) = 120





00

1122

33

44

-12-12

-02-020808

1818

2828

144144

004004064064

324324

784784

4040

2525

2020

0505

1010

576576

010010

128128

162162

784784



x x

TVs TVs

squaresquare

dd









f ( x )f ( x )

x x

UniformUniform

x x


x x


















11 andand x x 22

f ( x )f ( x )

x x

UniformUniform

x x 11 x x 11 x x 22 x x 22

x x


x x 11 x x 11 x x 22 x x 22

x x 11 x x 11 x x 22 x x 22


x x

f ( x )f ( x )

x x 11

x x 11

x x 22 x x 22

































wherewhere







E(E( x x ) = () = (aa ++ bb)2)2

= (5 + 15)2= (5 + 15)2

= 10= 10

E(E( x x ) = () = (aa ++ bb)2)2

= (5 + 15)2= (5 + 15)2

= 10= 10

Var(Var( x x ) = () = (bb -- aa))221212

= (15 ndash 5)= (15 ndash 5)22

1212= 833= 833

Var(Var( x x ) = () = (bb -- aa))221212

= (15 ndash 5)= (15 ndash 5)22

1212= 833= 833


n orm ro a





f ( x )f ( x )

x x 55 1010 1515

110110





f ( x )f ( x )

x x 55 1010 1515

110110


P(12 lt x lt 15) = (110)(3) = 3P(12 lt x lt 15) = (110)(3) = 3



saladsalad

1212












orma ro a y











orma ro a y




2 2( ) 21( )

2

x f x e micro σ

σ π

minus minus

=



π π = 314159= 314159

ee = 271828= 271828

wherewhere








x x




















x x





-10-10 00 2020





x x





σ = 15= 15

σ = 25= 25




x x













55 55

x x









6826682668266826









9972997299729972











68266826

95449544

99729972













σ σ = 1= 1

00

z z








z x

=

minus micro

σ









includingincluding


stock of stock of


order isorder is

placedplaced




are beingare being


It haIt ha




1515










= (20 - 15)6= (20 - 15)6= 83= 83


= (20 - 15)6= (20 - 15)6= 83= 83












= 05- 2967= 05- 2967

= 2033= 2033


= 2033= 2033








d d l b bili i ib i












00

Area = 9500Area = 9500

Area = 0500Area = 0500

z z z z

0505








= 15 + 1645(6)= 15 + 1645(6)

= 2487 or 25= 2487 or 25


= 15 + 1645(6)= 15 + 1645(6)

= 2487 or 25= 2487 or 25

























above



Characteristics







the n trials



Binomial Formula


( ) r N r r N r N

r

r N r

r N q pr N r


minus===

)(

)(


000

002

004

006

008

010

012

014

0 5 10 15 20 25 3k

P

( k

5 0

1 3 )

Expectation Value

micro = np = 7 13 = 2333

000

010

020

030

040

0 2 4 6 8 10k

P

( k

7

1 3 )







( )( ) (1 )

( ) x n x n

f x p p x n x

minus= minusminus





Binomial Formulas

Mean

Standard Deviation

( )

( ) ( )

( ) ( ) ( )

Np

p p N p


pm N pq pm

q p p

N

m

m N m N m

N

m

m N m N m

N

m

m N m N m

N

m

m N m N m

N

m

m N m N m

N

m

m N m N m

=




=

sum

minusminusminus

=

minusminus

=

minusminus

=

minusminus

=

minusminus

=

minusminus

=

minus

micro

micro micro

part

part

111

0

1

0

1

0

1

0

1

0

1

0

)1(1)1(

)1()1()1()1(

0)1)((

0

σ 2=


N

sum

P(m N p)m=0

N

sum

=Npq




( )( ) (1 )

( ) x n x n

f x p p x n x

minus= minusminus



FunctionFunction























next yearnext year








f xn

x n x p p x n x( )

( )( )( )

=

minus

minusminus1

1 23(1) (01) (09) 3(1)(81) 2431(3 1)f = = =

minus

LetLet p p = 10= 10 nn = 3= 3 x x = 1= 1




Tree Diagram



Leaves

(1)

Leaves

(1)

Stays(9)

Stays(9)

33

22

00

22

22



S (9)S (9)

Stays (9)Stays (9)

Stays (9)Stays (9)

S (9)S (9)

S (9)S (9)

S (9)S (9)

L (1)L (1)

L (1)L (1)

L (1)L (1)

L (1)L (1) 00100010

00900090

00900090

72907290

00900090

11

11

08100810

08100810

08100810

1111





n x 05 10 15 20 25 30 35 40 45 50

3 0 8574 7290 6141 5120 4219 3430 2746 2160 1664 1250

1 1354 2430 3251 3840 4219 4410 4436 4320 4084 3750

2 0071 0270 0574 0960 1406 1890 2389 2880 3341 3750

3 0001 0010 0034 0080 0156 0270 0429 0640 0911 1250

p






(1 )np pσ = minus


Var(Var( x x ) =) = σ σ 22



s ExpectedExpected

ValueValue

s VariancVarianc

ee

s StandardStandard

DeviationDeviation









s VarianceVariance




Applications







gets an offer





Answer

Selected = 03456+ 02592+ 00778

=06826

Not Selected = 1-06826 = 03174



Binomial Tables





Applications



















































f xe

x

x

( )

=

minus micro micro

wherewhere



ee = 271828= 271828








weekendweekend

eveningsevenings








4 33 (271828)(4) 1680

4f

minus

= =








000

005

010

015

020

025

0 1 2 3 4 5 6 7 8 9 10


P r o

b a

b i l i t

y



















Example













( )

xe

p x x

λ λ minus

=

45 0

45(0)0

e pminus

=

with λ =45

So p(0) = 001111



Similarly p(1)=004999 and p(2)=011248


is 001111+ 004999 + 011248 =017358















σ = minus(1 )np p








p yp y










Density Function

p y


f x e x( ) =

minus1

micro micro






P x x ex

( )le = minus minus

0 1 o micro

wherewhere









of of


thatthat


22





x x

f ( x )f ( x )

11

33

44

22


Distribution
























Le




















FamilyFamily

sizesize






Sell anSell an



DiscreteDiscrete






















ΣΣ f f (( x x ) = 1) = 1ΣΣ f f (( x x ) = 1) = 1





NumberNumber


11 5050

22 4040

33 101044 2020

200200

x x f f (( x x ))00 4040

11 2525

22 2020

33 050544 1010

100100

8020080200




1010

2020

3030

4040

5050

0 1 2 3 40 1 2 3 4


P r o

b a

b i l i t

y

P r o

b a

b i l i t

y

P r o

b a

b i l i t

y

P r o

b a

b i l i t

y



















Expected Value




x x f f (( x x )) xf xf (( x x ))

00 4040 0000

11 2525 2525

22 2020 4040

33 0505 1515

44 1010 4040

EE(( x x ) = 120) = 120





00

1122

33

44

-12-12

-02-020808

1818

2828

144144

004004064064

324324

784784

4040

2525

2020

0505

1010

576576

010010

128128

162162

784784



x x

TVs TVs

squaresquare

dd









f ( x )f ( x )

x x

UniformUniform

x x


x x


















11 andand x x 22

f ( x )f ( x )

x x

UniformUniform

x x 11 x x 11 x x 22 x x 22

x x


x x 11 x x 11 x x 22 x x 22

x x 11 x x 11 x x 22 x x 22


x x

f ( x )f ( x )

x x 11

x x 11

x x 22 x x 22

































wherewhere







E(E( x x ) = () = (aa ++ bb)2)2

= (5 + 15)2= (5 + 15)2

= 10= 10

E(E( x x ) = () = (aa ++ bb)2)2

= (5 + 15)2= (5 + 15)2

= 10= 10

Var(Var( x x ) = () = (bb -- aa))221212

= (15 ndash 5)= (15 ndash 5)22

1212= 833= 833

Var(Var( x x ) = () = (bb -- aa))221212

= (15 ndash 5)= (15 ndash 5)22

1212= 833= 833


n orm ro a





f ( x )f ( x )

x x 55 1010 1515

110110





f ( x )f ( x )

x x 55 1010 1515

110110


P(12 lt x lt 15) = (110)(3) = 3P(12 lt x lt 15) = (110)(3) = 3



saladsalad

1212












orma ro a y











orma ro a y




2 2( ) 21( )

2

x f x e micro σ

σ π

minus minus

=



π π = 314159= 314159

ee = 271828= 271828

wherewhere








x x




















x x





-10-10 00 2020





x x





σ = 15= 15

σ = 25= 25




x x













55 55

x x









6826682668266826









9972997299729972











68266826

95449544

99729972













σ σ = 1= 1

00

z z








z x

=

minus micro

σ









includingincluding


stock of stock of


order isorder is

placedplaced




are beingare being


It haIt ha




1515










= (20 - 15)6= (20 - 15)6= 83= 83


= (20 - 15)6= (20 - 15)6= 83= 83












= 05- 2967= 05- 2967

= 2033= 2033


= 2033= 2033








d d l b bili i ib i












00

Area = 9500Area = 9500

Area = 0500Area = 0500

z z z z

0505








= 15 + 1645(6)= 15 + 1645(6)

= 2487 or 25= 2487 or 25


= 15 + 1645(6)= 15 + 1645(6)

= 2487 or 25= 2487 or 25

























above



Characteristics







the n trials



Binomial Formula


( ) r N r r N r N

r

r N r

r N q pr N r


minus===

)(

)(


000

002

004

006

008

010

012

014

0 5 10 15 20 25 3k

P

( k

5 0

1 3 )

Expectation Value

micro = np = 7 13 = 2333

000

010

020

030

040

0 2 4 6 8 10k

P

( k

7

1 3 )







( )( ) (1 )

( ) x n x n

f x p p x n x

minus= minusminus





Binomial Formulas

Mean

Standard Deviation

( )

( ) ( )

( ) ( ) ( )

Np

p p N p


pm N pq pm

q p p

N

m

m N m N m

N

m

m N m N m

N

m

m N m N m

N

m

m N m N m

N

m

m N m N m

N

m

m N m N m

=




=

sum

minusminusminus

=

minusminus

=

minusminus

=

minusminus

=

minusminus

=

minusminus

=

minus

micro

micro micro

part

part

111

0

1

0

1

0

1

0

1

0

1

0

)1(1)1(

)1()1()1()1(

0)1)((

0

σ 2=


N

sum

P(m N p)m=0

N

sum

=Npq




( )( ) (1 )

( ) x n x n

f x p p x n x

minus= minusminus



FunctionFunction























next yearnext year








f xn

x n x p p x n x( )

( )( )( )

=

minus

minusminus1

1 23(1) (01) (09) 3(1)(81) 2431(3 1)f = = =

minus

LetLet p p = 10= 10 nn = 3= 3 x x = 1= 1




Tree Diagram



Leaves

(1)

Leaves

(1)

Stays(9)

Stays(9)

33

22

00

22

22



S (9)S (9)

Stays (9)Stays (9)

Stays (9)Stays (9)

S (9)S (9)

S (9)S (9)

S (9)S (9)

L (1)L (1)

L (1)L (1)

L (1)L (1)

L (1)L (1) 00100010

00900090

00900090

72907290

00900090

11

11

08100810

08100810

08100810

1111





n x 05 10 15 20 25 30 35 40 45 50

3 0 8574 7290 6141 5120 4219 3430 2746 2160 1664 1250

1 1354 2430 3251 3840 4219 4410 4436 4320 4084 3750

2 0071 0270 0574 0960 1406 1890 2389 2880 3341 3750

3 0001 0010 0034 0080 0156 0270 0429 0640 0911 1250

p






(1 )np pσ = minus


Var(Var( x x ) =) = σ σ 22



s ExpectedExpected

ValueValue

s VariancVarianc

ee

s StandardStandard

DeviationDeviation









s VarianceVariance




Applications







gets an offer





Answer

Selected = 03456+ 02592+ 00778

=06826

Not Selected = 1-06826 = 03174



Binomial Tables





Applications



















































f xe

x

x

( )

=

minus micro micro

wherewhere



ee = 271828= 271828








weekendweekend

eveningsevenings








4 33 (271828)(4) 1680

4f

minus

= =








000

005

010

015

020

025

0 1 2 3 4 5 6 7 8 9 10


P r o

b a

b i l i t

y



















Example













( )

xe

p x x

λ λ minus

=

45 0

45(0)0

e pminus

=

with λ =45

So p(0) = 001111



Similarly p(1)=004999 and p(2)=011248


is 001111+ 004999 + 011248 =017358















σ = minus(1 )np p








p yp y










Density Function

p y


f x e x( ) =

minus1

micro micro






P x x ex

( )le = minus minus

0 1 o micro

wherewhere









of of


thatthat


22





x x

f ( x )f ( x )

11

33

44

22


Distribution



































FamilyFamily

sizesize






Sell anSell an



DiscreteDiscrete






















ΣΣ f f (( x x ) = 1) = 1ΣΣ f f (( x x ) = 1) = 1





NumberNumber


11 5050

22 4040

33 101044 2020

200200

x x f f (( x x ))00 4040

11 2525

22 2020

33 050544 1010

100100

8020080200




1010

2020

3030

4040

5050

0 1 2 3 40 1 2 3 4


P r o

b a

b i l i t

y

P r o

b a

b i l i t

y

P r o

b a

b i l i t

y

P r o

b a

b i l i t

y



















Expected Value




x x f f (( x x )) xf xf (( x x ))

00 4040 0000

11 2525 2525

22 2020 4040

33 0505 1515

44 1010 4040

EE(( x x ) = 120) = 120





00

1122

33

44

-12-12

-02-020808

1818

2828

144144

004004064064

324324

784784

4040

2525

2020

0505

1010

576576

010010

128128

162162

784784



x x

TVs TVs

squaresquare

dd









f ( x )f ( x )

x x

UniformUniform

x x


x x


















11 andand x x 22

f ( x )f ( x )

x x

UniformUniform

x x 11 x x 11 x x 22 x x 22

x x


x x 11 x x 11 x x 22 x x 22

x x 11 x x 11 x x 22 x x 22


x x

f ( x )f ( x )

x x 11

x x 11

x x 22 x x 22

































wherewhere







E(E( x x ) = () = (aa ++ bb)2)2

= (5 + 15)2= (5 + 15)2

= 10= 10

E(E( x x ) = () = (aa ++ bb)2)2

= (5 + 15)2= (5 + 15)2

= 10= 10

Var(Var( x x ) = () = (bb -- aa))221212

= (15 ndash 5)= (15 ndash 5)22

1212= 833= 833

Var(Var( x x ) = () = (bb -- aa))221212

= (15 ndash 5)= (15 ndash 5)22

1212= 833= 833


n orm ro a





f ( x )f ( x )

x x 55 1010 1515

110110





f ( x )f ( x )

x x 55 1010 1515

110110


P(12 lt x lt 15) = (110)(3) = 3P(12 lt x lt 15) = (110)(3) = 3



saladsalad

1212












orma ro a y











orma ro a y




2 2( ) 21( )

2

x f x e micro σ

σ π

minus minus

=



π π = 314159= 314159

ee = 271828= 271828

wherewhere








x x




















x x





-10-10 00 2020





x x





σ = 15= 15

σ = 25= 25




x x













55 55

x x









6826682668266826









9972997299729972











68266826

95449544

99729972













σ σ = 1= 1

00

z z








z x

=

minus micro

σ









includingincluding


stock of stock of


order isorder is

placedplaced




are beingare being


It haIt ha




1515










= (20 - 15)6= (20 - 15)6= 83= 83


= (20 - 15)6= (20 - 15)6= 83= 83












= 05- 2967= 05- 2967

= 2033= 2033


= 2033= 2033








d d l b bili i ib i












00

Area = 9500Area = 9500

Area = 0500Area = 0500

z z z z

0505








= 15 + 1645(6)= 15 + 1645(6)

= 2487 or 25= 2487 or 25


= 15 + 1645(6)= 15 + 1645(6)

= 2487 or 25= 2487 or 25

























above



Characteristics







the n trials



Binomial Formula


( ) r N r r N r N

r

r N r

r N q pr N r


minus===

)(

)(


000

002

004

006

008

010

012

014

0 5 10 15 20 25 3k

P

( k

5 0

1 3 )

Expectation Value

micro = np = 7 13 = 2333

000

010

020

030

040

0 2 4 6 8 10k

P

( k

7

1 3 )







( )( ) (1 )

( ) x n x n

f x p p x n x

minus= minusminus





Binomial Formulas

Mean

Standard Deviation

( )

( ) ( )

( ) ( ) ( )

Np

p p N p


pm N pq pm

q p p

N

m

m N m N m

N

m

m N m N m

N

m

m N m N m

N

m

m N m N m

N

m

m N m N m

N

m

m N m N m

=




=

sum

minusminusminus

=

minusminus

=

minusminus

=

minusminus

=

minusminus

=

minusminus

=

minus

micro

micro micro

part

part

111

0

1

0

1

0

1

0

1

0

1

0

)1(1)1(

)1()1()1()1(

0)1)((

0

σ 2=


N

sum

P(m N p)m=0

N

sum

=Npq




( )( ) (1 )

( ) x n x n

f x p p x n x

minus= minusminus



FunctionFunction























next yearnext year








f xn

x n x p p x n x( )

( )( )( )

=

minus

minusminus1

1 23(1) (01) (09) 3(1)(81) 2431(3 1)f = = =

minus

LetLet p p = 10= 10 nn = 3= 3 x x = 1= 1




Tree Diagram



Leaves

(1)

Leaves

(1)

Stays(9)

Stays(9)

33

22

00

22

22



S (9)S (9)

Stays (9)Stays (9)

Stays (9)Stays (9)

S (9)S (9)

S (9)S (9)

S (9)S (9)

L (1)L (1)

L (1)L (1)

L (1)L (1)

L (1)L (1) 00100010

00900090

00900090

72907290

00900090

11

11

08100810

08100810

08100810

1111





n x 05 10 15 20 25 30 35 40 45 50

3 0 8574 7290 6141 5120 4219 3430 2746 2160 1664 1250

1 1354 2430 3251 3840 4219 4410 4436 4320 4084 3750

2 0071 0270 0574 0960 1406 1890 2389 2880 3341 3750

3 0001 0010 0034 0080 0156 0270 0429 0640 0911 1250

p






(1 )np pσ = minus


Var(Var( x x ) =) = σ σ 22



s ExpectedExpected

ValueValue

s VariancVarianc

ee

s StandardStandard

DeviationDeviation









s VarianceVariance




Applications







gets an offer





Answer

Selected = 03456+ 02592+ 00778

=06826

Not Selected = 1-06826 = 03174



Binomial Tables





Applications



















































f xe

x

x

( )

=

minus micro micro

wherewhere



ee = 271828= 271828








weekendweekend

eveningsevenings








4 33 (271828)(4) 1680

4f

minus

= =








000

005

010

015

020

025

0 1 2 3 4 5 6 7 8 9 10


P r o

b a

b i l i t

y



















Example













( )

xe

p x x

λ λ minus

=

45 0

45(0)0

e pminus

=

with λ =45

So p(0) = 001111



Similarly p(1)=004999 and p(2)=011248


is 001111+ 004999 + 011248 =017358















σ = minus(1 )np p








p yp y










Density Function

p y


f x e x( ) =

minus1

micro micro






P x x ex

( )le = minus minus

0 1 o micro

wherewhere









of of


thatthat


22





x x

f ( x )f ( x )

11

33

44

22


Distribution









































ΣΣ f f (( x x ) = 1) = 1ΣΣ f f (( x x ) = 1) = 1





NumberNumber


11 5050

22 4040

33 101044 2020

200200

x x f f (( x x ))00 4040

11 2525

22 2020

33 050544 1010

100100

8020080200




1010

2020

3030

4040

5050

0 1 2 3 40 1 2 3 4


P r o

b a

b i l i t

y

P r o

b a

b i l i t

y

P r o

b a

b i l i t

y

P r o

b a

b i l i t

y



















Expected Value




x x f f (( x x )) xf xf (( x x ))

00 4040 0000

11 2525 2525

22 2020 4040

33 0505 1515

44 1010 4040

EE(( x x ) = 120) = 120





00

1122

33

44

-12-12

-02-020808

1818

2828

144144

004004064064

324324

784784

4040

2525

2020

0505

1010

576576

010010

128128

162162

784784



x x

TVs TVs

squaresquare

dd









f ( x )f ( x )

x x

UniformUniform

x x


x x


















11 andand x x 22

f ( x )f ( x )

x x

UniformUniform

x x 11 x x 11 x x 22 x x 22

x x


x x 11 x x 11 x x 22 x x 22

x x 11 x x 11 x x 22 x x 22


x x

f ( x )f ( x )

x x 11

x x 11

x x 22 x x 22

































wherewhere







E(E( x x ) = () = (aa ++ bb)2)2

= (5 + 15)2= (5 + 15)2

= 10= 10

E(E( x x ) = () = (aa ++ bb)2)2

= (5 + 15)2= (5 + 15)2

= 10= 10

Var(Var( x x ) = () = (bb -- aa))221212

= (15 ndash 5)= (15 ndash 5)22

1212= 833= 833

Var(Var( x x ) = () = (bb -- aa))221212

= (15 ndash 5)= (15 ndash 5)22

1212= 833= 833


n orm ro a





f ( x )f ( x )

x x 55 1010 1515

110110





f ( x )f ( x )

x x 55 1010 1515

110110


P(12 lt x lt 15) = (110)(3) = 3P(12 lt x lt 15) = (110)(3) = 3



saladsalad

1212












orma ro a y











orma ro a y




2 2( ) 21( )

2

x f x e micro σ

σ π

minus minus

=



π π = 314159= 314159

ee = 271828= 271828

wherewhere








x x




















x x





-10-10 00 2020





x x





σ = 15= 15

σ = 25= 25




x x













55 55

x x









6826682668266826









9972997299729972











68266826

95449544

99729972













σ σ = 1= 1

00

z z








z x

=

minus micro

σ









includingincluding


stock of stock of


order isorder is

placedplaced




are beingare being


It haIt ha




1515










= (20 - 15)6= (20 - 15)6= 83= 83


= (20 - 15)6= (20 - 15)6= 83= 83












= 05- 2967= 05- 2967

= 2033= 2033


= 2033= 2033








d d l b bili i ib i












00

Area = 9500Area = 9500

Area = 0500Area = 0500

z z z z

0505








= 15 + 1645(6)= 15 + 1645(6)

= 2487 or 25= 2487 or 25


= 15 + 1645(6)= 15 + 1645(6)

= 2487 or 25= 2487 or 25

























above



Characteristics







the n trials



Binomial Formula


( ) r N r r N r N

r

r N r

r N q pr N r


minus===

)(

)(


000

002

004

006

008

010

012

014

0 5 10 15 20 25 3k

P

( k

5 0

1 3 )

Expectation Value

micro = np = 7 13 = 2333

000

010

020

030

040

0 2 4 6 8 10k

P

( k

7

1 3 )







( )( ) (1 )

( ) x n x n

f x p p x n x

minus= minusminus





Binomial Formulas

Mean

Standard Deviation

( )

( ) ( )

( ) ( ) ( )

Np

p p N p


pm N pq pm

q p p

N

m

m N m N m

N

m

m N m N m

N

m

m N m N m

N

m

m N m N m

N

m

m N m N m

N

m

m N m N m

=




=

sum

minusminusminus

=

minusminus

=

minusminus

=

minusminus

=

minusminus

=

minusminus

=

minus

micro

micro micro

part

part

111

0

1

0

1

0

1

0

1

0

1

0

)1(1)1(

)1()1()1()1(

0)1)((

0

σ 2=


N

sum

P(m N p)m=0

N

sum

=Npq




( )( ) (1 )

( ) x n x n

f x p p x n x

minus= minusminus



FunctionFunction























next yearnext year








f xn

x n x p p x n x( )

( )( )( )

=

minus

minusminus1

1 23(1) (01) (09) 3(1)(81) 2431(3 1)f = = =

minus

LetLet p p = 10= 10 nn = 3= 3 x x = 1= 1




Tree Diagram



Leaves

(1)

Leaves

(1)

Stays(9)

Stays(9)

33

22

00

22

22



S (9)S (9)

Stays (9)Stays (9)

Stays (9)Stays (9)

S (9)S (9)

S (9)S (9)

S (9)S (9)

L (1)L (1)

L (1)L (1)

L (1)L (1)

L (1)L (1) 00100010

00900090

00900090

72907290

00900090

11

11

08100810

08100810

08100810

1111





n x 05 10 15 20 25 30 35 40 45 50

3 0 8574 7290 6141 5120 4219 3430 2746 2160 1664 1250

1 1354 2430 3251 3840 4219 4410 4436 4320 4084 3750

2 0071 0270 0574 0960 1406 1890 2389 2880 3341 3750

3 0001 0010 0034 0080 0156 0270 0429 0640 0911 1250

p






(1 )np pσ = minus


Var(Var( x x ) =) = σ σ 22



s ExpectedExpected

ValueValue

s VariancVarianc

ee

s StandardStandard

DeviationDeviation









s VarianceVariance




Applications







gets an offer





Answer

Selected = 03456+ 02592+ 00778

=06826

Not Selected = 1-06826 = 03174



Binomial Tables





Applications



















































f xe

x

x

( )

=

minus micro micro

wherewhere



ee = 271828= 271828








weekendweekend

eveningsevenings








4 33 (271828)(4) 1680

4f

minus

= =








000

005

010

015

020

025

0 1 2 3 4 5 6 7 8 9 10


P r o

b a

b i l i t

y



















Example













( )

xe

p x x

λ λ minus

=

45 0

45(0)0

e pminus

=

with λ =45

So p(0) = 001111



Similarly p(1)=004999 and p(2)=011248


is 001111+ 004999 + 011248 =017358















σ = minus(1 )np p








p yp y










Density Function

p y


f x e x( ) =

minus1

micro micro






P x x ex

( )le = minus minus

0 1 o micro

wherewhere









of of


thatthat


22





x x

f ( x )f ( x )

11

33

44

22


Distribution
























NumberNumber


11 5050

22 4040

33 101044 2020

200200

x x f f (( x x ))00 4040

11 2525

22 2020

33 050544 1010

100100

8020080200




1010

2020

3030

4040

5050

0 1 2 3 40 1 2 3 4


P r o

b a

b i l i t

y

P r o

b a

b i l i t

y

P r o

b a

b i l i t

y

P r o

b a

b i l i t

y



















Expected Value




x x f f (( x x )) xf xf (( x x ))

00 4040 0000

11 2525 2525

22 2020 4040

33 0505 1515

44 1010 4040

EE(( x x ) = 120) = 120





00

1122

33

44

-12-12

-02-020808

1818

2828

144144

004004064064

324324

784784

4040

2525

2020

0505

1010

576576

010010

128128

162162

784784



x x

TVs TVs

squaresquare

dd









f ( x )f ( x )

x x

UniformUniform

x x


x x


















11 andand x x 22

f ( x )f ( x )

x x

UniformUniform

x x 11 x x 11 x x 22 x x 22

x x


x x 11 x x 11 x x 22 x x 22

x x 11 x x 11 x x 22 x x 22


x x

f ( x )f ( x )

x x 11

x x 11

x x 22 x x 22

































wherewhere







E(E( x x ) = () = (aa ++ bb)2)2

= (5 + 15)2= (5 + 15)2

= 10= 10

E(E( x x ) = () = (aa ++ bb)2)2

= (5 + 15)2= (5 + 15)2

= 10= 10

Var(Var( x x ) = () = (bb -- aa))221212

= (15 ndash 5)= (15 ndash 5)22

1212= 833= 833

Var(Var( x x ) = () = (bb -- aa))221212

= (15 ndash 5)= (15 ndash 5)22

1212= 833= 833


n orm ro a





f ( x )f ( x )

x x 55 1010 1515

110110





f ( x )f ( x )

x x 55 1010 1515

110110


P(12 lt x lt 15) = (110)(3) = 3P(12 lt x lt 15) = (110)(3) = 3



saladsalad

1212












orma ro a y











orma ro a y




2 2( ) 21( )

2

x f x e micro σ

σ π

minus minus

=



π π = 314159= 314159

ee = 271828= 271828

wherewhere








x x




















x x





-10-10 00 2020





x x





σ = 15= 15

σ = 25= 25




x x













55 55

x x









6826682668266826









9972997299729972











68266826

95449544

99729972













σ σ = 1= 1

00

z z








z x

=

minus micro

σ









includingincluding


stock of stock of


order isorder is

placedplaced




are beingare being


It haIt ha




1515










= (20 - 15)6= (20 - 15)6= 83= 83


= (20 - 15)6= (20 - 15)6= 83= 83












= 05- 2967= 05- 2967

= 2033= 2033


= 2033= 2033








d d l b bili i ib i












00

Area = 9500Area = 9500

Area = 0500Area = 0500

z z z z

0505








= 15 + 1645(6)= 15 + 1645(6)

= 2487 or 25= 2487 or 25


= 15 + 1645(6)= 15 + 1645(6)

= 2487 or 25= 2487 or 25

























above



Characteristics







the n trials



Binomial Formula


( ) r N r r N r N

r

r N r

r N q pr N r


minus===

)(

)(


000

002

004

006

008

010

012

014

0 5 10 15 20 25 3k

P

( k

5 0

1 3 )

Expectation Value

micro = np = 7 13 = 2333

000

010

020

030

040

0 2 4 6 8 10k

P

( k

7

1 3 )







( )( ) (1 )

( ) x n x n

f x p p x n x

minus= minusminus





Binomial Formulas

Mean

Standard Deviation

( )

( ) ( )

( ) ( ) ( )

Np

p p N p


pm N pq pm

q p p

N

m

m N m N m

N

m

m N m N m

N

m

m N m N m

N

m

m N m N m

N

m

m N m N m

N

m

m N m N m

=




=

sum

minusminusminus

=

minusminus

=

minusminus

=

minusminus

=

minusminus

=

minusminus

=

minus

micro

micro micro

part

part

111

0

1

0

1

0

1

0

1

0

1

0

)1(1)1(

)1()1()1()1(

0)1)((

0

σ 2=


N

sum

P(m N p)m=0

N

sum

=Npq




( )( ) (1 )

( ) x n x n

f x p p x n x

minus= minusminus



FunctionFunction























next yearnext year








f xn

x n x p p x n x( )

( )( )( )

=

minus

minusminus1

1 23(1) (01) (09) 3(1)(81) 2431(3 1)f = = =

minus

LetLet p p = 10= 10 nn = 3= 3 x x = 1= 1




Tree Diagram



Leaves

(1)

Leaves

(1)

Stays(9)

Stays(9)

33

22

00

22

22



S (9)S (9)

Stays (9)Stays (9)

Stays (9)Stays (9)

S (9)S (9)

S (9)S (9)

S (9)S (9)

L (1)L (1)

L (1)L (1)

L (1)L (1)

L (1)L (1) 00100010

00900090

00900090

72907290

00900090

11

11

08100810

08100810

08100810

1111





n x 05 10 15 20 25 30 35 40 45 50

3 0 8574 7290 6141 5120 4219 3430 2746 2160 1664 1250

1 1354 2430 3251 3840 4219 4410 4436 4320 4084 3750

2 0071 0270 0574 0960 1406 1890 2389 2880 3341 3750

3 0001 0010 0034 0080 0156 0270 0429 0640 0911 1250

p






(1 )np pσ = minus


Var(Var( x x ) =) = σ σ 22



s ExpectedExpected

ValueValue

s VariancVarianc

ee

s StandardStandard

DeviationDeviation









s VarianceVariance




Applications







gets an offer





Answer

Selected = 03456+ 02592+ 00778

=06826

Not Selected = 1-06826 = 03174



Binomial Tables





Applications



















































f xe

x

x

( )

=

minus micro micro

wherewhere



ee = 271828= 271828








weekendweekend

eveningsevenings








4 33 (271828)(4) 1680

4f

minus

= =








000

005

010

015

020

025

0 1 2 3 4 5 6 7 8 9 10


P r o

b a

b i l i t

y



















Example













( )

xe

p x x

λ λ minus

=

45 0

45(0)0

e pminus

=

with λ =45

So p(0) = 001111



Similarly p(1)=004999 and p(2)=011248


is 001111+ 004999 + 011248 =017358















σ = minus(1 )np p








p yp y










Density Function

p y


f x e x( ) =

minus1

micro micro






P x x ex

( )le = minus minus

0 1 o micro

wherewhere









of of


thatthat


22





x x

f ( x )f ( x )

11

33

44

22


Distribution






















1010

2020

3030

4040

5050

0 1 2 3 40 1 2 3 4


P r o

b a

b i l i t

y

P r o

b a

b i l i t

y

P r o

b a

b i l i t

y

P r o

b a

b i l i t

y



















Expected Value




x x f f (( x x )) xf xf (( x x ))

00 4040 0000

11 2525 2525

22 2020 4040

33 0505 1515

44 1010 4040

EE(( x x ) = 120) = 120





00

1122

33

44

-12-12

-02-020808

1818

2828

144144

004004064064

324324

784784

4040

2525

2020

0505

1010

576576

010010

128128

162162

784784



x x

TVs TVs

squaresquare

dd









f ( x )f ( x )

x x

UniformUniform

x x


x x


















11 andand x x 22

f ( x )f ( x )

x x

UniformUniform

x x 11 x x 11 x x 22 x x 22

x x


x x 11 x x 11 x x 22 x x 22

x x 11 x x 11 x x 22 x x 22


x x

f ( x )f ( x )

x x 11

x x 11

x x 22 x x 22

































wherewhere







E(E( x x ) = () = (aa ++ bb)2)2

= (5 + 15)2= (5 + 15)2

= 10= 10

E(E( x x ) = () = (aa ++ bb)2)2

= (5 + 15)2= (5 + 15)2

= 10= 10

Var(Var( x x ) = () = (bb -- aa))221212

= (15 ndash 5)= (15 ndash 5)22

1212= 833= 833

Var(Var( x x ) = () = (bb -- aa))221212

= (15 ndash 5)= (15 ndash 5)22

1212= 833= 833


n orm ro a





f ( x )f ( x )

x x 55 1010 1515

110110





f ( x )f ( x )

x x 55 1010 1515

110110


P(12 lt x lt 15) = (110)(3) = 3P(12 lt x lt 15) = (110)(3) = 3



saladsalad

1212












orma ro a y











orma ro a y




2 2( ) 21( )

2

x f x e micro σ

σ π

minus minus

=



π π = 314159= 314159

ee = 271828= 271828

wherewhere








x x




















x x





-10-10 00 2020





x x





σ = 15= 15

σ = 25= 25




x x













55 55

x x









6826682668266826









9972997299729972











68266826

95449544

99729972













σ σ = 1= 1

00

z z








z x

=

minus micro

σ









includingincluding


stock of stock of


order isorder is

placedplaced




are beingare being


It haIt ha




1515










= (20 - 15)6= (20 - 15)6= 83= 83


= (20 - 15)6= (20 - 15)6= 83= 83












= 05- 2967= 05- 2967

= 2033= 2033


= 2033= 2033








d d l b bili i ib i












00

Area = 9500Area = 9500

Area = 0500Area = 0500

z z z z

0505








= 15 + 1645(6)= 15 + 1645(6)

= 2487 or 25= 2487 or 25


= 15 + 1645(6)= 15 + 1645(6)

= 2487 or 25= 2487 or 25

























above



Characteristics







the n trials



Binomial Formula


( ) r N r r N r N

r

r N r

r N q pr N r


minus===

)(

)(


000

002

004

006

008

010

012

014

0 5 10 15 20 25 3k

P

( k

5 0

1 3 )

Expectation Value

micro = np = 7 13 = 2333

000

010

020

030

040

0 2 4 6 8 10k

P

( k

7

1 3 )







( )( ) (1 )

( ) x n x n

f x p p x n x

minus= minusminus





Binomial Formulas

Mean

Standard Deviation

( )

( ) ( )

( ) ( ) ( )

Np

p p N p


pm N pq pm

q p p

N

m

m N m N m

N

m

m N m N m

N

m

m N m N m

N

m

m N m N m

N

m

m N m N m

N

m

m N m N m

=




=

sum

minusminusminus

=

minusminus

=

minusminus

=

minusminus

=

minusminus

=

minusminus

=

minus

micro

micro micro

part

part

111

0

1

0

1

0

1

0

1

0

1

0

)1(1)1(

)1()1()1()1(

0)1)((

0

σ 2=


N

sum

P(m N p)m=0

N

sum

=Npq




( )( ) (1 )

( ) x n x n

f x p p x n x

minus= minusminus



FunctionFunction























next yearnext year








f xn

x n x p p x n x( )

( )( )( )

=

minus

minusminus1

1 23(1) (01) (09) 3(1)(81) 2431(3 1)f = = =

minus

LetLet p p = 10= 10 nn = 3= 3 x x = 1= 1




Tree Diagram



Leaves

(1)

Leaves

(1)

Stays(9)

Stays(9)

33

22

00

22

22



S (9)S (9)

Stays (9)Stays (9)

Stays (9)Stays (9)

S (9)S (9)

S (9)S (9)

S (9)S (9)

L (1)L (1)

L (1)L (1)

L (1)L (1)

L (1)L (1) 00100010

00900090

00900090

72907290

00900090

11

11

08100810

08100810

08100810

1111





n x 05 10 15 20 25 30 35 40 45 50

3 0 8574 7290 6141 5120 4219 3430 2746 2160 1664 1250

1 1354 2430 3251 3840 4219 4410 4436 4320 4084 3750

2 0071 0270 0574 0960 1406 1890 2389 2880 3341 3750

3 0001 0010 0034 0080 0156 0270 0429 0640 0911 1250

p






(1 )np pσ = minus


Var(Var( x x ) =) = σ σ 22



s ExpectedExpected

ValueValue

s VariancVarianc

ee

s StandardStandard

DeviationDeviation









s VarianceVariance




Applications







gets an offer





Answer

Selected = 03456+ 02592+ 00778

=06826

Not Selected = 1-06826 = 03174



Binomial Tables





Applications



















































f xe

x

x

( )

=

minus micro micro

wherewhere



ee = 271828= 271828








weekendweekend

eveningsevenings








4 33 (271828)(4) 1680

4f

minus

= =








000

005

010

015

020

025

0 1 2 3 4 5 6 7 8 9 10


P r o

b a

b i l i t

y



















Example













( )

xe

p x x

λ λ minus

=

45 0

45(0)0

e pminus

=

with λ =45

So p(0) = 001111



Similarly p(1)=004999 and p(2)=011248


is 001111+ 004999 + 011248 =017358















σ = minus(1 )np p








p yp y










Density Function

p y


f x e x( ) =

minus1

micro micro






P x x ex

( )le = minus minus

0 1 o micro

wherewhere









of of


thatthat


22





x x

f ( x )f ( x )

11

33

44

22


Distribution




































Expected Value




x x f f (( x x )) xf xf (( x x ))

00 4040 0000

11 2525 2525

22 2020 4040

33 0505 1515

44 1010 4040

EE(( x x ) = 120) = 120





00

1122

33

44

-12-12

-02-020808

1818

2828

144144

004004064064

324324

784784

4040

2525

2020

0505

1010

576576

010010

128128

162162

784784



x x

TVs TVs

squaresquare

dd









f ( x )f ( x )

x x

UniformUniform

x x


x x


















11 andand x x 22

f ( x )f ( x )

x x

UniformUniform

x x 11 x x 11 x x 22 x x 22

x x


x x 11 x x 11 x x 22 x x 22

x x 11 x x 11 x x 22 x x 22


x x

f ( x )f ( x )

x x 11

x x 11

x x 22 x x 22

































wherewhere







E(E( x x ) = () = (aa ++ bb)2)2

= (5 + 15)2= (5 + 15)2

= 10= 10

E(E( x x ) = () = (aa ++ bb)2)2

= (5 + 15)2= (5 + 15)2

= 10= 10

Var(Var( x x ) = () = (bb -- aa))221212

= (15 ndash 5)= (15 ndash 5)22

1212= 833= 833

Var(Var( x x ) = () = (bb -- aa))221212

= (15 ndash 5)= (15 ndash 5)22

1212= 833= 833


n orm ro a





f ( x )f ( x )

x x 55 1010 1515

110110





f ( x )f ( x )

x x 55 1010 1515

110110


P(12 lt x lt 15) = (110)(3) = 3P(12 lt x lt 15) = (110)(3) = 3



saladsalad

1212












orma ro a y











orma ro a y




2 2( ) 21( )

2

x f x e micro σ

σ π

minus minus

=



π π = 314159= 314159

ee = 271828= 271828

wherewhere








x x




















x x





-10-10 00 2020





x x





σ = 15= 15

σ = 25= 25




x x













55 55

x x









6826682668266826









9972997299729972











68266826

95449544

99729972













σ σ = 1= 1

00

z z








z x

=

minus micro

σ









includingincluding


stock of stock of


order isorder is

placedplaced




are beingare being


It haIt ha




1515










= (20 - 15)6= (20 - 15)6= 83= 83


= (20 - 15)6= (20 - 15)6= 83= 83












= 05- 2967= 05- 2967

= 2033= 2033


= 2033= 2033








d d l b bili i ib i












00

Area = 9500Area = 9500

Area = 0500Area = 0500

z z z z

0505








= 15 + 1645(6)= 15 + 1645(6)

= 2487 or 25= 2487 or 25


= 15 + 1645(6)= 15 + 1645(6)

= 2487 or 25= 2487 or 25

























above



Characteristics







the n trials



Binomial Formula


( ) r N r r N r N

r

r N r

r N q pr N r


minus===

)(

)(


000

002

004

006

008

010

012

014

0 5 10 15 20 25 3k

P

( k

5 0

1 3 )

Expectation Value

micro = np = 7 13 = 2333

000

010

020

030

040

0 2 4 6 8 10k

P

( k

7

1 3 )







( )( ) (1 )

( ) x n x n

f x p p x n x

minus= minusminus





Binomial Formulas

Mean

Standard Deviation

( )

( ) ( )

( ) ( ) ( )

Np

p p N p


pm N pq pm

q p p

N

m

m N m N m

N

m

m N m N m

N

m

m N m N m

N

m

m N m N m

N

m

m N m N m

N

m

m N m N m

=




=

sum

minusminusminus

=

minusminus

=

minusminus

=

minusminus

=

minusminus

=

minusminus

=

minus

micro

micro micro

part

part

111

0

1

0

1

0

1

0

1

0

1

0

)1(1)1(

)1()1()1()1(

0)1)((

0

σ 2=


N

sum

P(m N p)m=0

N

sum

=Npq




( )( ) (1 )

( ) x n x n

f x p p x n x

minus= minusminus



FunctionFunction























next yearnext year








f xn

x n x p p x n x( )

( )( )( )

=

minus

minusminus1

1 23(1) (01) (09) 3(1)(81) 2431(3 1)f = = =

minus

LetLet p p = 10= 10 nn = 3= 3 x x = 1= 1




Tree Diagram



Leaves

(1)

Leaves

(1)

Stays(9)

Stays(9)

33

22

00

22

22



S (9)S (9)

Stays (9)Stays (9)

Stays (9)Stays (9)

S (9)S (9)

S (9)S (9)

S (9)S (9)

L (1)L (1)

L (1)L (1)

L (1)L (1)

L (1)L (1) 00100010

00900090

00900090

72907290

00900090

11

11

08100810

08100810

08100810

1111





n x 05 10 15 20 25 30 35 40 45 50

3 0 8574 7290 6141 5120 4219 3430 2746 2160 1664 1250

1 1354 2430 3251 3840 4219 4410 4436 4320 4084 3750

2 0071 0270 0574 0960 1406 1890 2389 2880 3341 3750

3 0001 0010 0034 0080 0156 0270 0429 0640 0911 1250

p






(1 )np pσ = minus


Var(Var( x x ) =) = σ σ 22



s ExpectedExpected

ValueValue

s VariancVarianc

ee

s StandardStandard

DeviationDeviation









s VarianceVariance




Applications







gets an offer





Answer

Selected = 03456+ 02592+ 00778

=06826

Not Selected = 1-06826 = 03174



Binomial Tables





Applications



















































f xe

x

x

( )

=

minus micro micro

wherewhere



ee = 271828= 271828








weekendweekend

eveningsevenings








4 33 (271828)(4) 1680

4f

minus

= =








000

005

010

015

020

025

0 1 2 3 4 5 6 7 8 9 10


P r o

b a

b i l i t

y



















Example













( )

xe

p x x

λ λ minus

=

45 0

45(0)0

e pminus

=

with λ =45

So p(0) = 001111



Similarly p(1)=004999 and p(2)=011248


is 001111+ 004999 + 011248 =017358















σ = minus(1 )np p








p yp y










Density Function

p y


f x e x( ) =

minus1

micro micro






P x x ex

( )le = minus minus

0 1 o micro

wherewhere









of of


thatthat


22





x x

f ( x )f ( x )

11

33

44

22


Distribution























00

1122

33

44

-12-12

-02-020808

1818

2828

144144

004004064064

324324

784784

4040

2525

2020

0505

1010

576576

010010

128128

162162

784784



x x

TVs TVs

squaresquare

dd









f ( x )f ( x )

x x

UniformUniform

x x


x x


















11 andand x x 22

f ( x )f ( x )

x x

UniformUniform

x x 11 x x 11 x x 22 x x 22

x x


x x 11 x x 11 x x 22 x x 22

x x 11 x x 11 x x 22 x x 22


x x

f ( x )f ( x )

x x 11

x x 11

x x 22 x x 22

































wherewhere







E(E( x x ) = () = (aa ++ bb)2)2

= (5 + 15)2= (5 + 15)2

= 10= 10

E(E( x x ) = () = (aa ++ bb)2)2

= (5 + 15)2= (5 + 15)2

= 10= 10

Var(Var( x x ) = () = (bb -- aa))221212

= (15 ndash 5)= (15 ndash 5)22

1212= 833= 833

Var(Var( x x ) = () = (bb -- aa))221212

= (15 ndash 5)= (15 ndash 5)22

1212= 833= 833


n orm ro a





f ( x )f ( x )

x x 55 1010 1515

110110





f ( x )f ( x )

x x 55 1010 1515

110110


P(12 lt x lt 15) = (110)(3) = 3P(12 lt x lt 15) = (110)(3) = 3



saladsalad

1212












orma ro a y











orma ro a y




2 2( ) 21( )

2

x f x e micro σ

σ π

minus minus

=



π π = 314159= 314159

ee = 271828= 271828

wherewhere








x x




















x x





-10-10 00 2020





x x





σ = 15= 15

σ = 25= 25




x x













55 55

x x









6826682668266826









9972997299729972











68266826

95449544

99729972













σ σ = 1= 1

00

z z








z x

=

minus micro

σ









includingincluding


stock of stock of


order isorder is

placedplaced




are beingare being


It haIt ha




1515










= (20 - 15)6= (20 - 15)6= 83= 83


= (20 - 15)6= (20 - 15)6= 83= 83












= 05- 2967= 05- 2967

= 2033= 2033


= 2033= 2033








d d l b bili i ib i












00

Area = 9500Area = 9500

Area = 0500Area = 0500

z z z z

0505








= 15 + 1645(6)= 15 + 1645(6)

= 2487 or 25= 2487 or 25


= 15 + 1645(6)= 15 + 1645(6)

= 2487 or 25= 2487 or 25

























above



Characteristics







the n trials



Binomial Formula


( ) r N r r N r N

r

r N r

r N q pr N r


minus===

)(

)(


000

002

004

006

008

010

012

014

0 5 10 15 20 25 3k

P

( k

5 0

1 3 )

Expectation Value

micro = np = 7 13 = 2333

000

010

020

030

040

0 2 4 6 8 10k

P

( k

7

1 3 )







( )( ) (1 )

( ) x n x n

f x p p x n x

minus= minusminus





Binomial Formulas

Mean

Standard Deviation

( )

( ) ( )

( ) ( ) ( )

Np

p p N p


pm N pq pm

q p p

N

m

m N m N m

N

m

m N m N m

N

m

m N m N m

N

m

m N m N m

N

m

m N m N m

N

m

m N m N m

=




=

sum

minusminusminus

=

minusminus

=

minusminus

=

minusminus

=

minusminus

=

minusminus

=

minus

micro

micro micro

part

part

111

0

1

0

1

0

1

0

1

0

1

0

)1(1)1(

)1()1()1()1(

0)1)((

0

σ 2=


N

sum

P(m N p)m=0

N

sum

=Npq




( )( ) (1 )

( ) x n x n

f x p p x n x

minus= minusminus



FunctionFunction























next yearnext year








f xn

x n x p p x n x( )

( )( )( )

=

minus

minusminus1

1 23(1) (01) (09) 3(1)(81) 2431(3 1)f = = =

minus

LetLet p p = 10= 10 nn = 3= 3 x x = 1= 1




Tree Diagram



Leaves

(1)

Leaves

(1)

Stays(9)

Stays(9)

33

22

00

22

22



S (9)S (9)

Stays (9)Stays (9)

Stays (9)Stays (9)

S (9)S (9)

S (9)S (9)

S (9)S (9)

L (1)L (1)

L (1)L (1)

L (1)L (1)

L (1)L (1) 00100010

00900090

00900090

72907290

00900090

11

11

08100810

08100810

08100810

1111





n x 05 10 15 20 25 30 35 40 45 50

3 0 8574 7290 6141 5120 4219 3430 2746 2160 1664 1250

1 1354 2430 3251 3840 4219 4410 4436 4320 4084 3750

2 0071 0270 0574 0960 1406 1890 2389 2880 3341 3750

3 0001 0010 0034 0080 0156 0270 0429 0640 0911 1250

p






(1 )np pσ = minus


Var(Var( x x ) =) = σ σ 22



s ExpectedExpected

ValueValue

s VariancVarianc

ee

s StandardStandard

DeviationDeviation









s VarianceVariance




Applications







gets an offer





Answer

Selected = 03456+ 02592+ 00778

=06826

Not Selected = 1-06826 = 03174



Binomial Tables





Applications



















































f xe

x

x

( )

=

minus micro micro

wherewhere



ee = 271828= 271828








weekendweekend

eveningsevenings








4 33 (271828)(4) 1680

4f

minus

= =








000

005

010

015

020

025

0 1 2 3 4 5 6 7 8 9 10


P r o

b a

b i l i t

y



















Example













( )

xe

p x x

λ λ minus

=

45 0

45(0)0

e pminus

=

with λ =45

So p(0) = 001111



Similarly p(1)=004999 and p(2)=011248


is 001111+ 004999 + 011248 =017358















σ = minus(1 )np p








p yp y










Density Function

p y


f x e x( ) =

minus1

micro micro






P x x ex

( )le = minus minus

0 1 o micro

wherewhere









of of


thatthat


22





x x

f ( x )f ( x )

11

33

44

22


Distribution


























f ( x )f ( x )

x x

UniformUniform

x x


x x


















11 andand x x 22

f ( x )f ( x )

x x

UniformUniform

x x 11 x x 11 x x 22 x x 22

x x


x x 11 x x 11 x x 22 x x 22

x x 11 x x 11 x x 22 x x 22


x x

f ( x )f ( x )

x x 11

x x 11

x x 22 x x 22

































wherewhere







E(E( x x ) = () = (aa ++ bb)2)2

= (5 + 15)2= (5 + 15)2

= 10= 10

E(E( x x ) = () = (aa ++ bb)2)2

= (5 + 15)2= (5 + 15)2

= 10= 10

Var(Var( x x ) = () = (bb -- aa))221212

= (15 ndash 5)= (15 ndash 5)22

1212= 833= 833

Var(Var( x x ) = () = (bb -- aa))221212

= (15 ndash 5)= (15 ndash 5)22

1212= 833= 833


n orm ro a





f ( x )f ( x )

x x 55 1010 1515

110110





f ( x )f ( x )

x x 55 1010 1515

110110


P(12 lt x lt 15) = (110)(3) = 3P(12 lt x lt 15) = (110)(3) = 3



saladsalad

1212












orma ro a y











orma ro a y




2 2( ) 21( )

2

x f x e micro σ

σ π

minus minus

=



π π = 314159= 314159

ee = 271828= 271828

wherewhere








x x




















x x





-10-10 00 2020





x x





σ = 15= 15

σ = 25= 25




x x













55 55

x x









6826682668266826









9972997299729972











68266826

95449544

99729972













σ σ = 1= 1

00

z z








z x

=

minus micro

σ









includingincluding


stock of stock of


order isorder is

placedplaced




are beingare being


It haIt ha




1515










= (20 - 15)6= (20 - 15)6= 83= 83


= (20 - 15)6= (20 - 15)6= 83= 83












= 05- 2967= 05- 2967

= 2033= 2033


= 2033= 2033








d d l b bili i ib i












00

Area = 9500Area = 9500

Area = 0500Area = 0500

z z z z

0505








= 15 + 1645(6)= 15 + 1645(6)

= 2487 or 25= 2487 or 25


= 15 + 1645(6)= 15 + 1645(6)

= 2487 or 25= 2487 or 25

























above



Characteristics







the n trials



Binomial Formula


( ) r N r r N r N

r

r N r

r N q pr N r


minus===

)(

)(


000

002

004

006

008

010

012

014

0 5 10 15 20 25 3k

P

( k

5 0

1 3 )

Expectation Value

micro = np = 7 13 = 2333

000

010

020

030

040

0 2 4 6 8 10k

P

( k

7

1 3 )







( )( ) (1 )

( ) x n x n

f x p p x n x

minus= minusminus





Binomial Formulas

Mean

Standard Deviation

( )

( ) ( )

( ) ( ) ( )

Np

p p N p


pm N pq pm

q p p

N

m

m N m N m

N

m

m N m N m

N

m

m N m N m

N

m

m N m N m

N

m

m N m N m

N

m

m N m N m

=




=

sum

minusminusminus

=

minusminus

=

minusminus

=

minusminus

=

minusminus

=

minusminus

=

minus

micro

micro micro

part

part

111

0

1

0

1

0

1

0

1

0

1

0

)1(1)1(

)1()1()1()1(

0)1)((

0

σ 2=


N

sum

P(m N p)m=0

N

sum

=Npq




( )( ) (1 )

( ) x n x n

f x p p x n x

minus= minusminus



FunctionFunction























next yearnext year








f xn

x n x p p x n x( )

( )( )( )

=

minus

minusminus1

1 23(1) (01) (09) 3(1)(81) 2431(3 1)f = = =

minus

LetLet p p = 10= 10 nn = 3= 3 x x = 1= 1




Tree Diagram



Leaves

(1)

Leaves

(1)

Stays(9)

Stays(9)

33

22

00

22

22



S (9)S (9)

Stays (9)Stays (9)

Stays (9)Stays (9)

S (9)S (9)

S (9)S (9)

S (9)S (9)

L (1)L (1)

L (1)L (1)

L (1)L (1)

L (1)L (1) 00100010

00900090

00900090

72907290

00900090

11

11

08100810

08100810

08100810

1111





n x 05 10 15 20 25 30 35 40 45 50

3 0 8574 7290 6141 5120 4219 3430 2746 2160 1664 1250

1 1354 2430 3251 3840 4219 4410 4436 4320 4084 3750

2 0071 0270 0574 0960 1406 1890 2389 2880 3341 3750

3 0001 0010 0034 0080 0156 0270 0429 0640 0911 1250

p






(1 )np pσ = minus


Var(Var( x x ) =) = σ σ 22



s ExpectedExpected

ValueValue

s VariancVarianc

ee

s StandardStandard

DeviationDeviation









s VarianceVariance




Applications







gets an offer





Answer

Selected = 03456+ 02592+ 00778

=06826

Not Selected = 1-06826 = 03174



Binomial Tables





Applications



















































f xe

x

x

( )

=

minus micro micro

wherewhere



ee = 271828= 271828








weekendweekend

eveningsevenings








4 33 (271828)(4) 1680

4f

minus

= =








000

005

010

015

020

025

0 1 2 3 4 5 6 7 8 9 10


P r o

b a

b i l i t

y



















Example













( )

xe

p x x

λ λ minus

=

45 0

45(0)0

e pminus

=

with λ =45

So p(0) = 001111



Similarly p(1)=004999 and p(2)=011248


is 001111+ 004999 + 011248 =017358















σ = minus(1 )np p








p yp y










Density Function

p y


f x e x( ) =

minus1

micro micro






P x x ex

( )le = minus minus

0 1 o micro

wherewhere









of of


thatthat


22





x x

f ( x )f ( x )

11

33

44

22


Distribution




































11 andand x x 22

f ( x )f ( x )

x x

UniformUniform

x x 11 x x 11 x x 22 x x 22

x x


x x 11 x x 11 x x 22 x x 22

x x 11 x x 11 x x 22 x x 22


x x

f ( x )f ( x )

x x 11

x x 11

x x 22 x x 22

































wherewhere







E(E( x x ) = () = (aa ++ bb)2)2

= (5 + 15)2= (5 + 15)2

= 10= 10

E(E( x x ) = () = (aa ++ bb)2)2

= (5 + 15)2= (5 + 15)2

= 10= 10

Var(Var( x x ) = () = (bb -- aa))221212

= (15 ndash 5)= (15 ndash 5)22

1212= 833= 833

Var(Var( x x ) = () = (bb -- aa))221212

= (15 ndash 5)= (15 ndash 5)22

1212= 833= 833


n orm ro a





f ( x )f ( x )

x x 55 1010 1515

110110





f ( x )f ( x )

x x 55 1010 1515

110110


P(12 lt x lt 15) = (110)(3) = 3P(12 lt x lt 15) = (110)(3) = 3



saladsalad

1212












orma ro a y











orma ro a y




2 2( ) 21( )

2

x f x e micro σ

σ π

minus minus

=



π π = 314159= 314159

ee = 271828= 271828

wherewhere








x x




















x x





-10-10 00 2020





x x





σ = 15= 15

σ = 25= 25




x x













55 55

x x









6826682668266826









9972997299729972











68266826

95449544

99729972













σ σ = 1= 1

00

z z








z x

=

minus micro

σ









includingincluding


stock of stock of


order isorder is

placedplaced




are beingare being


It haIt ha




1515










= (20 - 15)6= (20 - 15)6= 83= 83


= (20 - 15)6= (20 - 15)6= 83= 83












= 05- 2967= 05- 2967

= 2033= 2033


= 2033= 2033








d d l b bili i ib i












00

Area = 9500Area = 9500

Area = 0500Area = 0500

z z z z

0505








= 15 + 1645(6)= 15 + 1645(6)

= 2487 or 25= 2487 or 25


= 15 + 1645(6)= 15 + 1645(6)

= 2487 or 25= 2487 or 25

























above



Characteristics







the n trials



Binomial Formula


( ) r N r r N r N

r

r N r

r N q pr N r


minus===

)(

)(


000

002

004

006

008

010

012

014

0 5 10 15 20 25 3k

P

( k

5 0

1 3 )

Expectation Value

micro = np = 7 13 = 2333

000

010

020

030

040

0 2 4 6 8 10k

P

( k

7

1 3 )







( )( ) (1 )

( ) x n x n

f x p p x n x

minus= minusminus





Binomial Formulas

Mean

Standard Deviation

( )

( ) ( )

( ) ( ) ( )

Np

p p N p


pm N pq pm

q p p

N

m

m N m N m

N

m

m N m N m

N

m

m N m N m

N

m

m N m N m

N

m

m N m N m

N

m

m N m N m

=




=

sum

minusminusminus

=

minusminus

=

minusminus

=

minusminus

=

minusminus

=

minusminus

=

minus

micro

micro micro

part

part

111

0

1

0

1

0

1

0

1

0

1

0

)1(1)1(

)1()1()1()1(

0)1)((

0

σ 2=


N

sum

P(m N p)m=0

N

sum

=Npq




( )( ) (1 )

( ) x n x n

f x p p x n x

minus= minusminus



FunctionFunction























next yearnext year








f xn

x n x p p x n x( )

( )( )( )

=

minus

minusminus1

1 23(1) (01) (09) 3(1)(81) 2431(3 1)f = = =

minus

LetLet p p = 10= 10 nn = 3= 3 x x = 1= 1




Tree Diagram



Leaves

(1)

Leaves

(1)

Stays(9)

Stays(9)

33

22

00

22

22



S (9)S (9)

Stays (9)Stays (9)

Stays (9)Stays (9)

S (9)S (9)

S (9)S (9)

S (9)S (9)

L (1)L (1)

L (1)L (1)

L (1)L (1)

L (1)L (1) 00100010

00900090

00900090

72907290

00900090

11

11

08100810

08100810

08100810

1111





n x 05 10 15 20 25 30 35 40 45 50

3 0 8574 7290 6141 5120 4219 3430 2746 2160 1664 1250

1 1354 2430 3251 3840 4219 4410 4436 4320 4084 3750

2 0071 0270 0574 0960 1406 1890 2389 2880 3341 3750

3 0001 0010 0034 0080 0156 0270 0429 0640 0911 1250

p






(1 )np pσ = minus


Var(Var( x x ) =) = σ σ 22



s ExpectedExpected

ValueValue

s VariancVarianc

ee

s StandardStandard

DeviationDeviation









s VarianceVariance




Applications







gets an offer





Answer

Selected = 03456+ 02592+ 00778

=06826

Not Selected = 1-06826 = 03174



Binomial Tables





Applications



















































f xe

x

x

( )

=

minus micro micro

wherewhere



ee = 271828= 271828








weekendweekend

eveningsevenings








4 33 (271828)(4) 1680

4f

minus

= =








000

005

010

015

020

025

0 1 2 3 4 5 6 7 8 9 10


P r o

b a

b i l i t

y



















Example













( )

xe

p x x

λ λ minus

=

45 0

45(0)0

e pminus

=

with λ =45

So p(0) = 001111



Similarly p(1)=004999 and p(2)=011248


is 001111+ 004999 + 011248 =017358















σ = minus(1 )np p








p yp y










Density Function

p y


f x e x( ) =

minus1

micro micro






P x x ex

( )le = minus minus

0 1 o micro

wherewhere









of of


thatthat


22





x x

f ( x )f ( x )

11

33

44

22


Distribution




















































wherewhere







E(E( x x ) = () = (aa ++ bb)2)2

= (5 + 15)2= (5 + 15)2

= 10= 10

E(E( x x ) = () = (aa ++ bb)2)2

= (5 + 15)2= (5 + 15)2

= 10= 10

Var(Var( x x ) = () = (bb -- aa))221212

= (15 ndash 5)= (15 ndash 5)22

1212= 833= 833

Var(Var( x x ) = () = (bb -- aa))221212

= (15 ndash 5)= (15 ndash 5)22

1212= 833= 833


n orm ro a





f ( x )f ( x )

x x 55 1010 1515

110110





f ( x )f ( x )

x x 55 1010 1515

110110


P(12 lt x lt 15) = (110)(3) = 3P(12 lt x lt 15) = (110)(3) = 3



saladsalad

1212












orma ro a y











orma ro a y




2 2( ) 21( )

2

x f x e micro σ

σ π

minus minus

=



π π = 314159= 314159

ee = 271828= 271828

wherewhere








x x




















x x





-10-10 00 2020





x x





σ = 15= 15

σ = 25= 25




x x













55 55

x x









6826682668266826









9972997299729972











68266826

95449544

99729972













σ σ = 1= 1

00

z z








z x

=

minus micro

σ









includingincluding


stock of stock of


order isorder is

placedplaced




are beingare being


It haIt ha




1515










= (20 - 15)6= (20 - 15)6= 83= 83


= (20 - 15)6= (20 - 15)6= 83= 83












= 05- 2967= 05- 2967

= 2033= 2033


= 2033= 2033








d d l b bili i ib i












00

Area = 9500Area = 9500

Area = 0500Area = 0500

z z z z

0505








= 15 + 1645(6)= 15 + 1645(6)

= 2487 or 25= 2487 or 25


= 15 + 1645(6)= 15 + 1645(6)

= 2487 or 25= 2487 or 25

























above



Characteristics







the n trials



Binomial Formula


( ) r N r r N r N

r

r N r

r N q pr N r


minus===

)(

)(


000

002

004

006

008

010

012

014

0 5 10 15 20 25 3k

P

( k

5 0

1 3 )

Expectation Value

micro = np = 7 13 = 2333

000

010

020

030

040

0 2 4 6 8 10k

P

( k

7

1 3 )







( )( ) (1 )

( ) x n x n

f x p p x n x

minus= minusminus





Binomial Formulas

Mean

Standard Deviation

( )

( ) ( )

( ) ( ) ( )

Np

p p N p


pm N pq pm

q p p

N

m

m N m N m

N

m

m N m N m

N

m

m N m N m

N

m

m N m N m

N

m

m N m N m

N

m

m N m N m

=




=

sum

minusminusminus

=

minusminus

=

minusminus

=

minusminus

=

minusminus

=

minusminus

=

minus

micro

micro micro

part

part

111

0

1

0

1

0

1

0

1

0

1

0

)1(1)1(

)1()1()1()1(

0)1)((

0

σ 2=


N

sum

P(m N p)m=0

N

sum

=Npq




( )( ) (1 )

( ) x n x n

f x p p x n x

minus= minusminus



FunctionFunction























next yearnext year








f xn

x n x p p x n x( )

( )( )( )

=

minus

minusminus1

1 23(1) (01) (09) 3(1)(81) 2431(3 1)f = = =

minus

LetLet p p = 10= 10 nn = 3= 3 x x = 1= 1




Tree Diagram



Leaves

(1)

Leaves

(1)

Stays(9)

Stays(9)

33

22

00

22

22



S (9)S (9)

Stays (9)Stays (9)

Stays (9)Stays (9)

S (9)S (9)

S (9)S (9)

S (9)S (9)

L (1)L (1)

L (1)L (1)

L (1)L (1)

L (1)L (1) 00100010

00900090

00900090

72907290

00900090

11

11

08100810

08100810

08100810

1111





n x 05 10 15 20 25 30 35 40 45 50

3 0 8574 7290 6141 5120 4219 3430 2746 2160 1664 1250

1 1354 2430 3251 3840 4219 4410 4436 4320 4084 3750

2 0071 0270 0574 0960 1406 1890 2389 2880 3341 3750

3 0001 0010 0034 0080 0156 0270 0429 0640 0911 1250

p






(1 )np pσ = minus


Var(Var( x x ) =) = σ σ 22



s ExpectedExpected

ValueValue

s VariancVarianc

ee

s StandardStandard

DeviationDeviation









s VarianceVariance




Applications







gets an offer





Answer

Selected = 03456+ 02592+ 00778

=06826

Not Selected = 1-06826 = 03174



Binomial Tables





Applications



















































f xe

x

x

( )

=

minus micro micro

wherewhere



ee = 271828= 271828








weekendweekend

eveningsevenings








4 33 (271828)(4) 1680

4f

minus

= =








000

005

010

015

020

025

0 1 2 3 4 5 6 7 8 9 10


P r o

b a

b i l i t

y



















Example













( )

xe

p x x

λ λ minus

=

45 0

45(0)0

e pminus

=

with λ =45

So p(0) = 001111



Similarly p(1)=004999 and p(2)=011248


is 001111+ 004999 + 011248 =017358















σ = minus(1 )np p








p yp y










Density Function

p y


f x e x( ) =

minus1

micro micro






P x x ex

( )le = minus minus

0 1 o micro

wherewhere









of of


thatthat


22





x x

f ( x )f ( x )

11

33

44

22


Distribution
























E(E( x x ) = () = (aa ++ bb)2)2

= (5 + 15)2= (5 + 15)2

= 10= 10

E(E( x x ) = () = (aa ++ bb)2)2

= (5 + 15)2= (5 + 15)2

= 10= 10

Var(Var( x x ) = () = (bb -- aa))221212

= (15 ndash 5)= (15 ndash 5)22

1212= 833= 833

Var(Var( x x ) = () = (bb -- aa))221212

= (15 ndash 5)= (15 ndash 5)22

1212= 833= 833


n orm ro a





f ( x )f ( x )

x x 55 1010 1515

110110





f ( x )f ( x )

x x 55 1010 1515

110110


P(12 lt x lt 15) = (110)(3) = 3P(12 lt x lt 15) = (110)(3) = 3



saladsalad

1212












orma ro a y











orma ro a y




2 2( ) 21( )

2

x f x e micro σ

σ π

minus minus

=



π π = 314159= 314159

ee = 271828= 271828

wherewhere








x x




















x x





-10-10 00 2020





x x





σ = 15= 15

σ = 25= 25




x x













55 55

x x









6826682668266826









9972997299729972











68266826

95449544

99729972













σ σ = 1= 1

00

z z








z x

=

minus micro

σ









includingincluding


stock of stock of


order isorder is

placedplaced




are beingare being


It haIt ha




1515










= (20 - 15)6= (20 - 15)6= 83= 83


= (20 - 15)6= (20 - 15)6= 83= 83












= 05- 2967= 05- 2967

= 2033= 2033


= 2033= 2033








d d l b bili i ib i












00

Area = 9500Area = 9500

Area = 0500Area = 0500

z z z z

0505








= 15 + 1645(6)= 15 + 1645(6)

= 2487 or 25= 2487 or 25


= 15 + 1645(6)= 15 + 1645(6)

= 2487 or 25= 2487 or 25

























above



Characteristics







the n trials



Binomial Formula


( ) r N r r N r N

r

r N r

r N q pr N r


minus===

)(

)(


000

002

004

006

008

010

012

014

0 5 10 15 20 25 3k

P

( k

5 0

1 3 )

Expectation Value

micro = np = 7 13 = 2333

000

010

020

030

040

0 2 4 6 8 10k

P

( k

7

1 3 )







( )( ) (1 )

( ) x n x n

f x p p x n x

minus= minusminus





Binomial Formulas

Mean

Standard Deviation

( )

( ) ( )

( ) ( ) ( )

Np

p p N p


pm N pq pm

q p p

N

m

m N m N m

N

m

m N m N m

N

m

m N m N m

N

m

m N m N m

N

m

m N m N m

N

m

m N m N m

=




=

sum

minusminusminus

=

minusminus

=

minusminus

=

minusminus

=

minusminus

=

minusminus

=

minus

micro

micro micro

part

part

111

0

1

0

1

0

1

0

1

0

1

0

)1(1)1(

)1()1()1()1(

0)1)((

0

σ 2=


N

sum

P(m N p)m=0

N

sum

=Npq




( )( ) (1 )

( ) x n x n

f x p p x n x

minus= minusminus



FunctionFunction























next yearnext year








f xn

x n x p p x n x( )

( )( )( )

=

minus

minusminus1

1 23(1) (01) (09) 3(1)(81) 2431(3 1)f = = =

minus

LetLet p p = 10= 10 nn = 3= 3 x x = 1= 1




Tree Diagram



Leaves

(1)

Leaves

(1)

Stays(9)

Stays(9)

33

22

00

22

22



S (9)S (9)

Stays (9)Stays (9)

Stays (9)Stays (9)

S (9)S (9)

S (9)S (9)

S (9)S (9)

L (1)L (1)

L (1)L (1)

L (1)L (1)

L (1)L (1) 00100010

00900090

00900090

72907290

00900090

11

11

08100810

08100810

08100810

1111





n x 05 10 15 20 25 30 35 40 45 50

3 0 8574 7290 6141 5120 4219 3430 2746 2160 1664 1250

1 1354 2430 3251 3840 4219 4410 4436 4320 4084 3750

2 0071 0270 0574 0960 1406 1890 2389 2880 3341 3750

3 0001 0010 0034 0080 0156 0270 0429 0640 0911 1250

p






(1 )np pσ = minus


Var(Var( x x ) =) = σ σ 22



s ExpectedExpected

ValueValue

s VariancVarianc

ee

s StandardStandard

DeviationDeviation









s VarianceVariance




Applications







gets an offer





Answer

Selected = 03456+ 02592+ 00778

=06826

Not Selected = 1-06826 = 03174



Binomial Tables





Applications



















































f xe

x

x

( )

=

minus micro micro

wherewhere



ee = 271828= 271828








weekendweekend

eveningsevenings








4 33 (271828)(4) 1680

4f

minus

= =








000

005

010

015

020

025

0 1 2 3 4 5 6 7 8 9 10


P r o

b a

b i l i t

y



















Example













( )

xe

p x x

λ λ minus

=

45 0

45(0)0

e pminus

=

with λ =45

So p(0) = 001111



Similarly p(1)=004999 and p(2)=011248


is 001111+ 004999 + 011248 =017358















σ = minus(1 )np p








p yp y










Density Function

p y


f x e x( ) =

minus1

micro micro






P x x ex

( )le = minus minus

0 1 o micro

wherewhere









of of


thatthat


22





x x

f ( x )f ( x )

11

33

44

22


Distribution
























f ( x )f ( x )

x x 55 1010 1515

110110





f ( x )f ( x )

x x 55 1010 1515

110110


P(12 lt x lt 15) = (110)(3) = 3P(12 lt x lt 15) = (110)(3) = 3



saladsalad

1212












orma ro a y











orma ro a y




2 2( ) 21( )

2

x f x e micro σ

σ π

minus minus

=



π π = 314159= 314159

ee = 271828= 271828

wherewhere








x x




















x x





-10-10 00 2020





x x





σ = 15= 15

σ = 25= 25




x x













55 55

x x









6826682668266826









9972997299729972











68266826

95449544

99729972













σ σ = 1= 1

00

z z








z x

=

minus micro

σ









includingincluding


stock of stock of


order isorder is

placedplaced




are beingare being


It haIt ha




1515










= (20 - 15)6= (20 - 15)6= 83= 83


= (20 - 15)6= (20 - 15)6= 83= 83












= 05- 2967= 05- 2967

= 2033= 2033


= 2033= 2033








d d l b bili i ib i












00

Area = 9500Area = 9500

Area = 0500Area = 0500

z z z z

0505








= 15 + 1645(6)= 15 + 1645(6)

= 2487 or 25= 2487 or 25


= 15 + 1645(6)= 15 + 1645(6)

= 2487 or 25= 2487 or 25

























above



Characteristics







the n trials



Binomial Formula


( ) r N r r N r N

r

r N r

r N q pr N r


minus===

)(

)(


000

002

004

006

008

010

012

014

0 5 10 15 20 25 3k

P

( k

5 0

1 3 )

Expectation Value

micro = np = 7 13 = 2333

000

010

020

030

040

0 2 4 6 8 10k

P

( k

7

1 3 )







( )( ) (1 )

( ) x n x n

f x p p x n x

minus= minusminus





Binomial Formulas

Mean

Standard Deviation

( )

( ) ( )

( ) ( ) ( )

Np

p p N p


pm N pq pm

q p p

N

m

m N m N m

N

m

m N m N m

N

m

m N m N m

N

m

m N m N m

N

m

m N m N m

N

m

m N m N m

=




=

sum

minusminusminus

=

minusminus

=

minusminus

=

minusminus

=

minusminus

=

minusminus

=

minus

micro

micro micro

part

part

111

0

1

0

1

0

1

0

1

0

1

0

)1(1)1(

)1()1()1()1(

0)1)((

0

σ 2=


N

sum

P(m N p)m=0

N

sum

=Npq




( )( ) (1 )

( ) x n x n

f x p p x n x

minus= minusminus



FunctionFunction























next yearnext year








f xn

x n x p p x n x( )

( )( )( )

=

minus

minusminus1

1 23(1) (01) (09) 3(1)(81) 2431(3 1)f = = =

minus

LetLet p p = 10= 10 nn = 3= 3 x x = 1= 1




Tree Diagram



Leaves

(1)

Leaves

(1)

Stays(9)

Stays(9)

33

22

00

22

22



S (9)S (9)

Stays (9)Stays (9)

Stays (9)Stays (9)

S (9)S (9)

S (9)S (9)

S (9)S (9)

L (1)L (1)

L (1)L (1)

L (1)L (1)

L (1)L (1) 00100010

00900090

00900090

72907290

00900090

11

11

08100810

08100810

08100810

1111





n x 05 10 15 20 25 30 35 40 45 50

3 0 8574 7290 6141 5120 4219 3430 2746 2160 1664 1250

1 1354 2430 3251 3840 4219 4410 4436 4320 4084 3750

2 0071 0270 0574 0960 1406 1890 2389 2880 3341 3750

3 0001 0010 0034 0080 0156 0270 0429 0640 0911 1250

p






(1 )np pσ = minus


Var(Var( x x ) =) = σ σ 22



s ExpectedExpected

ValueValue

s VariancVarianc

ee

s StandardStandard

DeviationDeviation









s VarianceVariance




Applications







gets an offer





Answer

Selected = 03456+ 02592+ 00778

=06826

Not Selected = 1-06826 = 03174



Binomial Tables





Applications



















































f xe

x

x

( )

=

minus micro micro

wherewhere



ee = 271828= 271828








weekendweekend

eveningsevenings








4 33 (271828)(4) 1680

4f

minus

= =








000

005

010

015

020

025

0 1 2 3 4 5 6 7 8 9 10


P r o

b a

b i l i t

y



















Example













( )

xe

p x x

λ λ minus

=

45 0

45(0)0

e pminus

=

with λ =45

So p(0) = 001111



Similarly p(1)=004999 and p(2)=011248


is 001111+ 004999 + 011248 =017358















σ = minus(1 )np p








p yp y










Density Function

p y


f x e x( ) =

minus1

micro micro






P x x ex

( )le = minus minus

0 1 o micro

wherewhere









of of


thatthat


22





x x

f ( x )f ( x )

11

33

44

22


Distribution






















f ( x )f ( x )

x x 55 1010 1515

110110


P(12 lt x lt 15) = (110)(3) = 3P(12 lt x lt 15) = (110)(3) = 3



saladsalad

1212












orma ro a y











orma ro a y




2 2( ) 21( )

2

x f x e micro σ

σ π

minus minus

=



π π = 314159= 314159

ee = 271828= 271828

wherewhere








x x




















x x





-10-10 00 2020





x x





σ = 15= 15

σ = 25= 25




x x













55 55

x x









6826682668266826









9972997299729972











68266826

95449544

99729972













σ σ = 1= 1

00

z z








z x

=

minus micro

σ









includingincluding


stock of stock of


order isorder is

placedplaced




are beingare being


It haIt ha




1515










= (20 - 15)6= (20 - 15)6= 83= 83


= (20 - 15)6= (20 - 15)6= 83= 83












= 05- 2967= 05- 2967

= 2033= 2033


= 2033= 2033








d d l b bili i ib i












00

Area = 9500Area = 9500

Area = 0500Area = 0500

z z z z

0505








= 15 + 1645(6)= 15 + 1645(6)

= 2487 or 25= 2487 or 25


= 15 + 1645(6)= 15 + 1645(6)

= 2487 or 25= 2487 or 25

























above



Characteristics







the n trials



Binomial Formula


( ) r N r r N r N

r

r N r

r N q pr N r


minus===

)(

)(


000

002

004

006

008

010

012

014

0 5 10 15 20 25 3k

P

( k

5 0

1 3 )

Expectation Value

micro = np = 7 13 = 2333

000

010

020

030

040

0 2 4 6 8 10k

P

( k

7

1 3 )







( )( ) (1 )

( ) x n x n

f x p p x n x

minus= minusminus





Binomial Formulas

Mean

Standard Deviation

( )

( ) ( )

( ) ( ) ( )

Np

p p N p


pm N pq pm

q p p

N

m

m N m N m

N

m

m N m N m

N

m

m N m N m

N

m

m N m N m

N

m

m N m N m

N

m

m N m N m

=




=

sum

minusminusminus

=

minusminus

=

minusminus

=

minusminus

=

minusminus

=

minusminus

=

minus

micro

micro micro

part

part

111

0

1

0

1

0

1

0

1

0

1

0

)1(1)1(

)1()1()1()1(

0)1)((

0

σ 2=


N

sum

P(m N p)m=0

N

sum

=Npq




( )( ) (1 )

( ) x n x n

f x p p x n x

minus= minusminus



FunctionFunction























next yearnext year








f xn

x n x p p x n x( )

( )( )( )

=

minus

minusminus1

1 23(1) (01) (09) 3(1)(81) 2431(3 1)f = = =

minus

LetLet p p = 10= 10 nn = 3= 3 x x = 1= 1




Tree Diagram



Leaves

(1)

Leaves

(1)

Stays(9)

Stays(9)

33

22

00

22

22



S (9)S (9)

Stays (9)Stays (9)

Stays (9)Stays (9)

S (9)S (9)

S (9)S (9)

S (9)S (9)

L (1)L (1)

L (1)L (1)

L (1)L (1)

L (1)L (1) 00100010

00900090

00900090

72907290

00900090

11

11

08100810

08100810

08100810

1111





n x 05 10 15 20 25 30 35 40 45 50

3 0 8574 7290 6141 5120 4219 3430 2746 2160 1664 1250

1 1354 2430 3251 3840 4219 4410 4436 4320 4084 3750

2 0071 0270 0574 0960 1406 1890 2389 2880 3341 3750

3 0001 0010 0034 0080 0156 0270 0429 0640 0911 1250

p






(1 )np pσ = minus


Var(Var( x x ) =) = σ σ 22



s ExpectedExpected

ValueValue

s VariancVarianc

ee

s StandardStandard

DeviationDeviation









s VarianceVariance




Applications







gets an offer





Answer

Selected = 03456+ 02592+ 00778

=06826

Not Selected = 1-06826 = 03174



Binomial Tables





Applications



















































f xe

x

x

( )

=

minus micro micro

wherewhere



ee = 271828= 271828








weekendweekend

eveningsevenings








4 33 (271828)(4) 1680

4f

minus

= =








000

005

010

015

020

025

0 1 2 3 4 5 6 7 8 9 10


P r o

b a

b i l i t

y



















Example













( )

xe

p x x

λ λ minus

=

45 0

45(0)0

e pminus

=

with λ =45

So p(0) = 001111



Similarly p(1)=004999 and p(2)=011248


is 001111+ 004999 + 011248 =017358















σ = minus(1 )np p








p yp y










Density Function

p y


f x e x( ) =

minus1

micro micro






P x x ex

( )le = minus minus

0 1 o micro

wherewhere









of of


thatthat


22





x x

f ( x )f ( x )

11

33

44

22


Distribution






























orma ro a y











orma ro a y




2 2( ) 21( )

2

x f x e micro σ

σ π

minus minus

=



π π = 314159= 314159

ee = 271828= 271828

wherewhere








x x




















x x





-10-10 00 2020





x x





σ = 15= 15

σ = 25= 25




x x













55 55

x x









6826682668266826









9972997299729972











68266826

95449544

99729972













σ σ = 1= 1

00

z z








z x

=

minus micro

σ









includingincluding


stock of stock of


order isorder is

placedplaced




are beingare being


It haIt ha




1515










= (20 - 15)6= (20 - 15)6= 83= 83


= (20 - 15)6= (20 - 15)6= 83= 83












= 05- 2967= 05- 2967

= 2033= 2033


= 2033= 2033








d d l b bili i ib i












00

Area = 9500Area = 9500

Area = 0500Area = 0500

z z z z

0505








= 15 + 1645(6)= 15 + 1645(6)

= 2487 or 25= 2487 or 25


= 15 + 1645(6)= 15 + 1645(6)

= 2487 or 25= 2487 or 25

























above



Characteristics







the n trials



Binomial Formula


( ) r N r r N r N

r

r N r

r N q pr N r


minus===

)(

)(


000

002

004

006

008

010

012

014

0 5 10 15 20 25 3k

P

( k

5 0

1 3 )

Expectation Value

micro = np = 7 13 = 2333

000

010

020

030

040

0 2 4 6 8 10k

P

( k

7

1 3 )







( )( ) (1 )

( ) x n x n

f x p p x n x

minus= minusminus





Binomial Formulas

Mean

Standard Deviation

( )

( ) ( )

( ) ( ) ( )

Np

p p N p


pm N pq pm

q p p

N

m

m N m N m

N

m

m N m N m

N

m

m N m N m

N

m

m N m N m

N

m

m N m N m

N

m

m N m N m

=




=

sum

minusminusminus

=

minusminus

=

minusminus

=

minusminus

=

minusminus

=

minusminus

=

minus

micro

micro micro

part

part

111

0

1

0

1

0

1

0

1

0

1

0

)1(1)1(

)1()1()1()1(

0)1)((

0

σ 2=


N

sum

P(m N p)m=0

N

sum

=Npq




( )( ) (1 )

( ) x n x n

f x p p x n x

minus= minusminus



FunctionFunction























next yearnext year








f xn

x n x p p x n x( )

( )( )( )

=

minus

minusminus1

1 23(1) (01) (09) 3(1)(81) 2431(3 1)f = = =

minus

LetLet p p = 10= 10 nn = 3= 3 x x = 1= 1




Tree Diagram



Leaves

(1)

Leaves

(1)

Stays(9)

Stays(9)

33

22

00

22

22



S (9)S (9)

Stays (9)Stays (9)

Stays (9)Stays (9)

S (9)S (9)

S (9)S (9)

S (9)S (9)

L (1)L (1)

L (1)L (1)

L (1)L (1)

L (1)L (1) 00100010

00900090

00900090

72907290

00900090

11

11

08100810

08100810

08100810

1111





n x 05 10 15 20 25 30 35 40 45 50

3 0 8574 7290 6141 5120 4219 3430 2746 2160 1664 1250

1 1354 2430 3251 3840 4219 4410 4436 4320 4084 3750

2 0071 0270 0574 0960 1406 1890 2389 2880 3341 3750

3 0001 0010 0034 0080 0156 0270 0429 0640 0911 1250

p






(1 )np pσ = minus


Var(Var( x x ) =) = σ σ 22



s ExpectedExpected

ValueValue

s VariancVarianc

ee

s StandardStandard

DeviationDeviation









s VarianceVariance




Applications







gets an offer





Answer

Selected = 03456+ 02592+ 00778

=06826

Not Selected = 1-06826 = 03174



Binomial Tables





Applications



















































f xe

x

x

( )

=

minus micro micro

wherewhere



ee = 271828= 271828








weekendweekend

eveningsevenings








4 33 (271828)(4) 1680

4f

minus

= =








000

005

010

015

020

025

0 1 2 3 4 5 6 7 8 9 10


P r o

b a

b i l i t

y



















Example













( )

xe

p x x

λ λ minus

=

45 0

45(0)0

e pminus

=

with λ =45

So p(0) = 001111



Similarly p(1)=004999 and p(2)=011248


is 001111+ 004999 + 011248 =017358















σ = minus(1 )np p








p yp y










Density Function

p y


f x e x( ) =

minus1

micro micro






P x x ex

( )le = minus minus

0 1 o micro

wherewhere









of of


thatthat


22





x x

f ( x )f ( x )

11

33

44

22


Distribution






























orma ro a y




2 2( ) 21( )

2

x f x e micro σ

σ π

minus minus

=



π π = 314159= 314159

ee = 271828= 271828

wherewhere








x x




















x x





-10-10 00 2020





x x





σ = 15= 15

σ = 25= 25




x x













55 55

x x









6826682668266826









9972997299729972











68266826

95449544

99729972













σ σ = 1= 1

00

z z








z x

=

minus micro

σ









includingincluding


stock of stock of


order isorder is

placedplaced




are beingare being


It haIt ha




1515










= (20 - 15)6= (20 - 15)6= 83= 83


= (20 - 15)6= (20 - 15)6= 83= 83












= 05- 2967= 05- 2967

= 2033= 2033


= 2033= 2033








d d l b bili i ib i












00

Area = 9500Area = 9500

Area = 0500Area = 0500

z z z z

0505








= 15 + 1645(6)= 15 + 1645(6)

= 2487 or 25= 2487 or 25


= 15 + 1645(6)= 15 + 1645(6)

= 2487 or 25= 2487 or 25

























above



Characteristics







the n trials



Binomial Formula


( ) r N r r N r N

r

r N r

r N q pr N r


minus===

)(

)(


000

002

004

006

008

010

012

014

0 5 10 15 20 25 3k

P

( k

5 0

1 3 )

Expectation Value

micro = np = 7 13 = 2333

000

010

020

030

040

0 2 4 6 8 10k

P

( k

7

1 3 )







( )( ) (1 )

( ) x n x n

f x p p x n x

minus= minusminus





Binomial Formulas

Mean

Standard Deviation

( )

( ) ( )

( ) ( ) ( )

Np

p p N p


pm N pq pm

q p p

N

m

m N m N m

N

m

m N m N m

N

m

m N m N m

N

m

m N m N m

N

m

m N m N m

N

m

m N m N m

=




=

sum

minusminusminus

=

minusminus

=

minusminus

=

minusminus

=

minusminus

=

minusminus

=

minus

micro

micro micro

part

part

111

0

1

0

1

0

1

0

1

0

1

0

)1(1)1(

)1()1()1()1(

0)1)((

0

σ 2=


N

sum

P(m N p)m=0

N

sum

=Npq




( )( ) (1 )

( ) x n x n

f x p p x n x

minus= minusminus



FunctionFunction























next yearnext year








f xn

x n x p p x n x( )

( )( )( )

=

minus

minusminus1

1 23(1) (01) (09) 3(1)(81) 2431(3 1)f = = =

minus

LetLet p p = 10= 10 nn = 3= 3 x x = 1= 1




Tree Diagram



Leaves

(1)

Leaves

(1)

Stays(9)

Stays(9)

33

22

00

22

22



S (9)S (9)

Stays (9)Stays (9)

Stays (9)Stays (9)

S (9)S (9)

S (9)S (9)

S (9)S (9)

L (1)L (1)

L (1)L (1)

L (1)L (1)

L (1)L (1) 00100010

00900090

00900090

72907290

00900090

11

11

08100810

08100810

08100810

1111





n x 05 10 15 20 25 30 35 40 45 50

3 0 8574 7290 6141 5120 4219 3430 2746 2160 1664 1250

1 1354 2430 3251 3840 4219 4410 4436 4320 4084 3750

2 0071 0270 0574 0960 1406 1890 2389 2880 3341 3750

3 0001 0010 0034 0080 0156 0270 0429 0640 0911 1250

p






(1 )np pσ = minus


Var(Var( x x ) =) = σ σ 22



s ExpectedExpected

ValueValue

s VariancVarianc

ee

s StandardStandard

DeviationDeviation









s VarianceVariance




Applications







gets an offer





Answer

Selected = 03456+ 02592+ 00778

=06826

Not Selected = 1-06826 = 03174



Binomial Tables





Applications



















































f xe

x

x

( )

=

minus micro micro

wherewhere



ee = 271828= 271828








weekendweekend

eveningsevenings








4 33 (271828)(4) 1680

4f

minus

= =








000

005

010

015

020

025

0 1 2 3 4 5 6 7 8 9 10


P r o

b a

b i l i t

y



















Example













( )

xe

p x x

λ λ minus

=

45 0

45(0)0

e pminus

=

with λ =45

So p(0) = 001111



Similarly p(1)=004999 and p(2)=011248


is 001111+ 004999 + 011248 =017358















σ = minus(1 )np p








p yp y










Density Function

p y


f x e x( ) =

minus1

micro micro






P x x ex

( )le = minus minus

0 1 o micro

wherewhere









of of


thatthat


22





x x

f ( x )f ( x )

11

33

44

22


Distribution























2 2( ) 21( )

2

x f x e micro σ

σ π

minus minus

=



π π = 314159= 314159

ee = 271828= 271828

wherewhere








x x




















x x





-10-10 00 2020





x x





σ = 15= 15

σ = 25= 25




x x













55 55

x x









6826682668266826









9972997299729972











68266826

95449544

99729972













σ σ = 1= 1

00

z z








z x

=

minus micro

σ









includingincluding


stock of stock of


order isorder is

placedplaced




are beingare being


It haIt ha




1515










= (20 - 15)6= (20 - 15)6= 83= 83


= (20 - 15)6= (20 - 15)6= 83= 83












= 05- 2967= 05- 2967

= 2033= 2033


= 2033= 2033








d d l b bili i ib i












00

Area = 9500Area = 9500

Area = 0500Area = 0500

z z z z

0505








= 15 + 1645(6)= 15 + 1645(6)

= 2487 or 25= 2487 or 25


= 15 + 1645(6)= 15 + 1645(6)

= 2487 or 25= 2487 or 25

























above



Characteristics







the n trials



Binomial Formula


( ) r N r r N r N

r

r N r

r N q pr N r


minus===

)(

)(


000

002

004

006

008

010

012

014

0 5 10 15 20 25 3k

P

( k

5 0

1 3 )

Expectation Value

micro = np = 7 13 = 2333

000

010

020

030

040

0 2 4 6 8 10k

P

( k

7

1 3 )







( )( ) (1 )

( ) x n x n

f x p p x n x

minus= minusminus





Binomial Formulas

Mean

Standard Deviation

( )

( ) ( )

( ) ( ) ( )

Np

p p N p


pm N pq pm

q p p

N

m

m N m N m

N

m

m N m N m

N

m

m N m N m

N

m

m N m N m

N

m

m N m N m

N

m

m N m N m

=




=

sum

minusminusminus

=

minusminus

=

minusminus

=

minusminus

=

minusminus

=

minusminus

=

minus

micro

micro micro

part

part

111

0

1

0

1

0

1

0

1

0

1

0

)1(1)1(

)1()1()1()1(

0)1)((

0

σ 2=


N

sum

P(m N p)m=0

N

sum

=Npq




( )( ) (1 )

( ) x n x n

f x p p x n x

minus= minusminus



FunctionFunction























next yearnext year








f xn

x n x p p x n x( )

( )( )( )

=

minus

minusminus1

1 23(1) (01) (09) 3(1)(81) 2431(3 1)f = = =

minus

LetLet p p = 10= 10 nn = 3= 3 x x = 1= 1




Tree Diagram



Leaves

(1)

Leaves

(1)

Stays(9)

Stays(9)

33

22

00

22

22



S (9)S (9)

Stays (9)Stays (9)

Stays (9)Stays (9)

S (9)S (9)

S (9)S (9)

S (9)S (9)

L (1)L (1)

L (1)L (1)

L (1)L (1)

L (1)L (1) 00100010

00900090

00900090

72907290

00900090

11

11

08100810

08100810

08100810

1111





n x 05 10 15 20 25 30 35 40 45 50

3 0 8574 7290 6141 5120 4219 3430 2746 2160 1664 1250

1 1354 2430 3251 3840 4219 4410 4436 4320 4084 3750

2 0071 0270 0574 0960 1406 1890 2389 2880 3341 3750

3 0001 0010 0034 0080 0156 0270 0429 0640 0911 1250

p






(1 )np pσ = minus


Var(Var( x x ) =) = σ σ 22



s ExpectedExpected

ValueValue

s VariancVarianc

ee

s StandardStandard

DeviationDeviation









s VarianceVariance




Applications







gets an offer





Answer

Selected = 03456+ 02592+ 00778

=06826

Not Selected = 1-06826 = 03174



Binomial Tables





Applications



















































f xe

x

x

( )

=

minus micro micro

wherewhere



ee = 271828= 271828








weekendweekend

eveningsevenings








4 33 (271828)(4) 1680

4f

minus

= =








000

005

010

015

020

025

0 1 2 3 4 5 6 7 8 9 10


P r o

b a

b i l i t

y



















Example













( )

xe

p x x

λ λ minus

=

45 0

45(0)0

e pminus

=

with λ =45

So p(0) = 001111



Similarly p(1)=004999 and p(2)=011248


is 001111+ 004999 + 011248 =017358















σ = minus(1 )np p








p yp y










Density Function

p y


f x e x( ) =

minus1

micro micro






P x x ex

( )le = minus minus

0 1 o micro

wherewhere









of of


thatthat


22





x x

f ( x )f ( x )

11

33

44

22


Distribution



























x x




















x x





-10-10 00 2020





x x





σ = 15= 15

σ = 25= 25




x x













55 55

x x









6826682668266826









9972997299729972











68266826

95449544

99729972













σ σ = 1= 1

00

z z








z x

=

minus micro

σ









includingincluding


stock of stock of


order isorder is

placedplaced




are beingare being


It haIt ha




1515










= (20 - 15)6= (20 - 15)6= 83= 83


= (20 - 15)6= (20 - 15)6= 83= 83












= 05- 2967= 05- 2967

= 2033= 2033


= 2033= 2033








d d l b bili i ib i












00

Area = 9500Area = 9500

Area = 0500Area = 0500

z z z z

0505








= 15 + 1645(6)= 15 + 1645(6)

= 2487 or 25= 2487 or 25


= 15 + 1645(6)= 15 + 1645(6)

= 2487 or 25= 2487 or 25

























above



Characteristics







the n trials



Binomial Formula


( ) r N r r N r N

r

r N r

r N q pr N r


minus===

)(

)(


000

002

004

006

008

010

012

014

0 5 10 15 20 25 3k

P

( k

5 0

1 3 )

Expectation Value

micro = np = 7 13 = 2333

000

010

020

030

040

0 2 4 6 8 10k

P

( k

7

1 3 )







( )( ) (1 )

( ) x n x n

f x p p x n x

minus= minusminus





Binomial Formulas

Mean

Standard Deviation

( )

( ) ( )

( ) ( ) ( )

Np

p p N p


pm N pq pm

q p p

N

m

m N m N m

N

m

m N m N m

N

m

m N m N m

N

m

m N m N m

N

m

m N m N m

N

m

m N m N m

=




=

sum

minusminusminus

=

minusminus

=

minusminus

=

minusminus

=

minusminus

=

minusminus

=

minus

micro

micro micro

part

part

111

0

1

0

1

0

1

0

1

0

1

0

)1(1)1(

)1()1()1()1(

0)1)((

0

σ 2=


N

sum

P(m N p)m=0

N

sum

=Npq




( )( ) (1 )

( ) x n x n

f x p p x n x

minus= minusminus



FunctionFunction























next yearnext year








f xn

x n x p p x n x( )

( )( )( )

=

minus

minusminus1

1 23(1) (01) (09) 3(1)(81) 2431(3 1)f = = =

minus

LetLet p p = 10= 10 nn = 3= 3 x x = 1= 1




Tree Diagram



Leaves

(1)

Leaves

(1)

Stays(9)

Stays(9)

33

22

00

22

22



S (9)S (9)

Stays (9)Stays (9)

Stays (9)Stays (9)

S (9)S (9)

S (9)S (9)

S (9)S (9)

L (1)L (1)

L (1)L (1)

L (1)L (1)

L (1)L (1) 00100010

00900090

00900090

72907290

00900090

11

11

08100810

08100810

08100810

1111





n x 05 10 15 20 25 30 35 40 45 50

3 0 8574 7290 6141 5120 4219 3430 2746 2160 1664 1250

1 1354 2430 3251 3840 4219 4410 4436 4320 4084 3750

2 0071 0270 0574 0960 1406 1890 2389 2880 3341 3750

3 0001 0010 0034 0080 0156 0270 0429 0640 0911 1250

p






(1 )np pσ = minus


Var(Var( x x ) =) = σ σ 22



s ExpectedExpected

ValueValue

s VariancVarianc

ee

s StandardStandard

DeviationDeviation









s VarianceVariance




Applications







gets an offer





Answer

Selected = 03456+ 02592+ 00778

=06826

Not Selected = 1-06826 = 03174



Binomial Tables





Applications



















































f xe

x

x

( )

=

minus micro micro

wherewhere



ee = 271828= 271828








weekendweekend

eveningsevenings








4 33 (271828)(4) 1680

4f

minus

= =








000

005

010

015

020

025

0 1 2 3 4 5 6 7 8 9 10


P r o

b a

b i l i t

y



















Example













( )

xe

p x x

λ λ minus

=

45 0

45(0)0

e pminus

=

with λ =45

So p(0) = 001111



Similarly p(1)=004999 and p(2)=011248


is 001111+ 004999 + 011248 =017358















σ = minus(1 )np p








p yp y










Density Function

p y


f x e x( ) =

minus1

micro micro






P x x ex

( )le = minus minus

0 1 o micro

wherewhere









of of


thatthat


22





x x

f ( x )f ( x )

11

33

44

22


Distribution







































x x





-10-10 00 2020





x x





σ = 15= 15

σ = 25= 25




x x













55 55

x x









6826682668266826









9972997299729972











68266826

95449544

99729972













σ σ = 1= 1

00

z z








z x

=

minus micro

σ









includingincluding


stock of stock of


order isorder is

placedplaced




are beingare being


It haIt ha




1515










= (20 - 15)6= (20 - 15)6= 83= 83


= (20 - 15)6= (20 - 15)6= 83= 83












= 05- 2967= 05- 2967

= 2033= 2033


= 2033= 2033








d d l b bili i ib i












00

Area = 9500Area = 9500

Area = 0500Area = 0500

z z z z

0505








= 15 + 1645(6)= 15 + 1645(6)

= 2487 or 25= 2487 or 25


= 15 + 1645(6)= 15 + 1645(6)

= 2487 or 25= 2487 or 25

























above



Characteristics







the n trials



Binomial Formula


( ) r N r r N r N

r

r N r

r N q pr N r


minus===

)(

)(


000

002

004

006

008

010

012

014

0 5 10 15 20 25 3k

P

( k

5 0

1 3 )

Expectation Value

micro = np = 7 13 = 2333

000

010

020

030

040

0 2 4 6 8 10k

P

( k

7

1 3 )







( )( ) (1 )

( ) x n x n

f x p p x n x

minus= minusminus





Binomial Formulas

Mean

Standard Deviation

( )

( ) ( )

( ) ( ) ( )

Np

p p N p


pm N pq pm

q p p

N

m

m N m N m

N

m

m N m N m

N

m

m N m N m

N

m

m N m N m

N

m

m N m N m

N

m

m N m N m

=




=

sum

minusminusminus

=

minusminus

=

minusminus

=

minusminus

=

minusminus

=

minusminus

=

minus

micro

micro micro

part

part

111

0

1

0

1

0

1

0

1

0

1

0

)1(1)1(

)1()1()1()1(

0)1)((

0

σ 2=


N

sum

P(m N p)m=0

N

sum

=Npq




( )( ) (1 )

( ) x n x n

f x p p x n x

minus= minusminus



FunctionFunction























next yearnext year








f xn

x n x p p x n x( )

( )( )( )

=

minus

minusminus1

1 23(1) (01) (09) 3(1)(81) 2431(3 1)f = = =

minus

LetLet p p = 10= 10 nn = 3= 3 x x = 1= 1




Tree Diagram



Leaves

(1)

Leaves

(1)

Stays(9)

Stays(9)

33

22

00

22

22



S (9)S (9)

Stays (9)Stays (9)

Stays (9)Stays (9)

S (9)S (9)

S (9)S (9)

S (9)S (9)

L (1)L (1)

L (1)L (1)

L (1)L (1)

L (1)L (1) 00100010

00900090

00900090

72907290

00900090

11

11

08100810

08100810

08100810

1111





n x 05 10 15 20 25 30 35 40 45 50

3 0 8574 7290 6141 5120 4219 3430 2746 2160 1664 1250

1 1354 2430 3251 3840 4219 4410 4436 4320 4084 3750

2 0071 0270 0574 0960 1406 1890 2389 2880 3341 3750

3 0001 0010 0034 0080 0156 0270 0429 0640 0911 1250

p






(1 )np pσ = minus


Var(Var( x x ) =) = σ σ 22



s ExpectedExpected

ValueValue

s VariancVarianc

ee

s StandardStandard

DeviationDeviation









s VarianceVariance




Applications







gets an offer





Answer

Selected = 03456+ 02592+ 00778

=06826

Not Selected = 1-06826 = 03174



Binomial Tables





Applications



















































f xe

x

x

( )

=

minus micro micro

wherewhere



ee = 271828= 271828








weekendweekend

eveningsevenings








4 33 (271828)(4) 1680

4f

minus

= =








000

005

010

015

020

025

0 1 2 3 4 5 6 7 8 9 10


P r o

b a

b i l i t

y



















Example













( )

xe

p x x

λ λ minus

=

45 0

45(0)0

e pminus

=

with λ =45

So p(0) = 001111



Similarly p(1)=004999 and p(2)=011248


is 001111+ 004999 + 011248 =017358















σ = minus(1 )np p








p yp y










Density Function

p y


f x e x( ) =

minus1

micro micro






P x x ex

( )le = minus minus

0 1 o micro

wherewhere









of of


thatthat


22





x x

f ( x )f ( x )

11

33

44

22


Distribution
























-10-10 00 2020





x x





σ = 15= 15

σ = 25= 25




x x













55 55

x x









6826682668266826









9972997299729972











68266826

95449544

99729972













σ σ = 1= 1

00

z z








z x

=

minus micro

σ









includingincluding


stock of stock of


order isorder is

placedplaced




are beingare being


It haIt ha




1515










= (20 - 15)6= (20 - 15)6= 83= 83


= (20 - 15)6= (20 - 15)6= 83= 83












= 05- 2967= 05- 2967

= 2033= 2033


= 2033= 2033








d d l b bili i ib i












00

Area = 9500Area = 9500

Area = 0500Area = 0500

z z z z

0505








= 15 + 1645(6)= 15 + 1645(6)

= 2487 or 25= 2487 or 25


= 15 + 1645(6)= 15 + 1645(6)

= 2487 or 25= 2487 or 25

























above



Characteristics







the n trials



Binomial Formula


( ) r N r r N r N

r

r N r

r N q pr N r


minus===

)(

)(


000

002

004

006

008

010

012

014

0 5 10 15 20 25 3k

P

( k

5 0

1 3 )

Expectation Value

micro = np = 7 13 = 2333

000

010

020

030

040

0 2 4 6 8 10k

P

( k

7

1 3 )







( )( ) (1 )

( ) x n x n

f x p p x n x

minus= minusminus





Binomial Formulas

Mean

Standard Deviation

( )

( ) ( )

( ) ( ) ( )

Np

p p N p


pm N pq pm

q p p

N

m

m N m N m

N

m

m N m N m

N

m

m N m N m

N

m

m N m N m

N

m

m N m N m

N

m

m N m N m

=




=

sum

minusminusminus

=

minusminus

=

minusminus

=

minusminus

=

minusminus

=

minusminus

=

minus

micro

micro micro

part

part

111

0

1

0

1

0

1

0

1

0

1

0

)1(1)1(

)1()1()1()1(

0)1)((

0

σ 2=


N

sum

P(m N p)m=0

N

sum

=Npq




( )( ) (1 )

( ) x n x n

f x p p x n x

minus= minusminus



FunctionFunction























next yearnext year








f xn

x n x p p x n x( )

( )( )( )

=

minus

minusminus1

1 23(1) (01) (09) 3(1)(81) 2431(3 1)f = = =

minus

LetLet p p = 10= 10 nn = 3= 3 x x = 1= 1




Tree Diagram



Leaves

(1)

Leaves

(1)

Stays(9)

Stays(9)

33

22

00

22

22



S (9)S (9)

Stays (9)Stays (9)

Stays (9)Stays (9)

S (9)S (9)

S (9)S (9)

S (9)S (9)

L (1)L (1)

L (1)L (1)

L (1)L (1)

L (1)L (1) 00100010

00900090

00900090

72907290

00900090

11

11

08100810

08100810

08100810

1111





n x 05 10 15 20 25 30 35 40 45 50

3 0 8574 7290 6141 5120 4219 3430 2746 2160 1664 1250

1 1354 2430 3251 3840 4219 4410 4436 4320 4084 3750

2 0071 0270 0574 0960 1406 1890 2389 2880 3341 3750

3 0001 0010 0034 0080 0156 0270 0429 0640 0911 1250

p






(1 )np pσ = minus


Var(Var( x x ) =) = σ σ 22



s ExpectedExpected

ValueValue

s VariancVarianc

ee

s StandardStandard

DeviationDeviation









s VarianceVariance




Applications







gets an offer





Answer

Selected = 03456+ 02592+ 00778

=06826

Not Selected = 1-06826 = 03174



Binomial Tables





Applications



















































f xe

x

x

( )

=

minus micro micro

wherewhere



ee = 271828= 271828








weekendweekend

eveningsevenings








4 33 (271828)(4) 1680

4f

minus

= =








000

005

010

015

020

025

0 1 2 3 4 5 6 7 8 9 10


P r o

b a

b i l i t

y



















Example













( )

xe

p x x

λ λ minus

=

45 0

45(0)0

e pminus

=

with λ =45

So p(0) = 001111



Similarly p(1)=004999 and p(2)=011248


is 001111+ 004999 + 011248 =017358















σ = minus(1 )np p








p yp y










Density Function

p y


f x e x( ) =

minus1

micro micro






P x x ex

( )le = minus minus

0 1 o micro

wherewhere









of of


thatthat


22





x x

f ( x )f ( x )

11

33

44

22


Distribution
























σ = 15= 15

σ = 25= 25




x x













55 55

x x









6826682668266826









9972997299729972











68266826

95449544

99729972













σ σ = 1= 1

00

z z








z x

=

minus micro

σ









includingincluding


stock of stock of


order isorder is

placedplaced




are beingare being


It haIt ha




1515










= (20 - 15)6= (20 - 15)6= 83= 83


= (20 - 15)6= (20 - 15)6= 83= 83












= 05- 2967= 05- 2967

= 2033= 2033


= 2033= 2033








d d l b bili i ib i












00

Area = 9500Area = 9500

Area = 0500Area = 0500

z z z z

0505








= 15 + 1645(6)= 15 + 1645(6)

= 2487 or 25= 2487 or 25


= 15 + 1645(6)= 15 + 1645(6)

= 2487 or 25= 2487 or 25

























above



Characteristics







the n trials



Binomial Formula


( ) r N r r N r N

r

r N r

r N q pr N r


minus===

)(

)(


000

002

004

006

008

010

012

014

0 5 10 15 20 25 3k

P

( k

5 0

1 3 )

Expectation Value

micro = np = 7 13 = 2333

000

010

020

030

040

0 2 4 6 8 10k

P

( k

7

1 3 )







( )( ) (1 )

( ) x n x n

f x p p x n x

minus= minusminus





Binomial Formulas

Mean

Standard Deviation

( )

( ) ( )

( ) ( ) ( )

Np

p p N p


pm N pq pm

q p p

N

m

m N m N m

N

m

m N m N m

N

m

m N m N m

N

m

m N m N m

N

m

m N m N m

N

m

m N m N m

=




=

sum

minusminusminus

=

minusminus

=

minusminus

=

minusminus

=

minusminus

=

minusminus

=

minus

micro

micro micro

part

part

111

0

1

0

1

0

1

0

1

0

1

0

)1(1)1(

)1()1()1()1(

0)1)((

0

σ 2=


N

sum

P(m N p)m=0

N

sum

=Npq




( )( ) (1 )

( ) x n x n

f x p p x n x

minus= minusminus



FunctionFunction























next yearnext year








f xn

x n x p p x n x( )

( )( )( )

=

minus

minusminus1

1 23(1) (01) (09) 3(1)(81) 2431(3 1)f = = =

minus

LetLet p p = 10= 10 nn = 3= 3 x x = 1= 1




Tree Diagram



Leaves

(1)

Leaves

(1)

Stays(9)

Stays(9)

33

22

00

22

22



S (9)S (9)

Stays (9)Stays (9)

Stays (9)Stays (9)

S (9)S (9)

S (9)S (9)

S (9)S (9)

L (1)L (1)

L (1)L (1)

L (1)L (1)

L (1)L (1) 00100010

00900090

00900090

72907290

00900090

11

11

08100810

08100810

08100810

1111





n x 05 10 15 20 25 30 35 40 45 50

3 0 8574 7290 6141 5120 4219 3430 2746 2160 1664 1250

1 1354 2430 3251 3840 4219 4410 4436 4320 4084 3750

2 0071 0270 0574 0960 1406 1890 2389 2880 3341 3750

3 0001 0010 0034 0080 0156 0270 0429 0640 0911 1250

p






(1 )np pσ = minus


Var(Var( x x ) =) = σ σ 22



s ExpectedExpected

ValueValue

s VariancVarianc

ee

s StandardStandard

DeviationDeviation









s VarianceVariance




Applications







gets an offer





Answer

Selected = 03456+ 02592+ 00778

=06826

Not Selected = 1-06826 = 03174



Binomial Tables





Applications



















































f xe

x

x

( )

=

minus micro micro

wherewhere



ee = 271828= 271828








weekendweekend

eveningsevenings








4 33 (271828)(4) 1680

4f

minus

= =








000

005

010

015

020

025

0 1 2 3 4 5 6 7 8 9 10


P r o

b a

b i l i t

y



















Example













( )

xe

p x x

λ λ minus

=

45 0

45(0)0

e pminus

=

with λ =45

So p(0) = 001111



Similarly p(1)=004999 and p(2)=011248


is 001111+ 004999 + 011248 =017358















σ = minus(1 )np p








p yp y










Density Function

p y


f x e x( ) =

minus1

micro micro






P x x ex

( )le = minus minus

0 1 o micro

wherewhere









of of


thatthat


22





x x

f ( x )f ( x )

11

33

44

22


Distribution
































55 55

x x









6826682668266826









9972997299729972











68266826

95449544

99729972













σ σ = 1= 1

00

z z








z x

=

minus micro

σ









includingincluding


stock of stock of


order isorder is

placedplaced




are beingare being


It haIt ha




1515










= (20 - 15)6= (20 - 15)6= 83= 83


= (20 - 15)6= (20 - 15)6= 83= 83












= 05- 2967= 05- 2967

= 2033= 2033


= 2033= 2033








d d l b bili i ib i












00

Area = 9500Area = 9500

Area = 0500Area = 0500

z z z z

0505








= 15 + 1645(6)= 15 + 1645(6)

= 2487 or 25= 2487 or 25


= 15 + 1645(6)= 15 + 1645(6)

= 2487 or 25= 2487 or 25

























above



Characteristics







the n trials



Binomial Formula


( ) r N r r N r N

r

r N r

r N q pr N r


minus===

)(

)(


000

002

004

006

008

010

012

014

0 5 10 15 20 25 3k

P

( k

5 0

1 3 )

Expectation Value

micro = np = 7 13 = 2333

000

010

020

030

040

0 2 4 6 8 10k

P

( k

7

1 3 )







( )( ) (1 )

( ) x n x n

f x p p x n x

minus= minusminus





Binomial Formulas

Mean

Standard Deviation

( )

( ) ( )

( ) ( ) ( )

Np

p p N p


pm N pq pm

q p p

N

m

m N m N m

N

m

m N m N m

N

m

m N m N m

N

m

m N m N m

N

m

m N m N m

N

m

m N m N m

=




=

sum

minusminusminus

=

minusminus

=

minusminus

=

minusminus

=

minusminus

=

minusminus

=

minus

micro

micro micro

part

part

111

0

1

0

1

0

1

0

1

0

1

0

)1(1)1(

)1()1()1()1(

0)1)((

0

σ 2=


N

sum

P(m N p)m=0

N

sum

=Npq




( )( ) (1 )

( ) x n x n

f x p p x n x

minus= minusminus



FunctionFunction























next yearnext year








f xn

x n x p p x n x( )

( )( )( )

=

minus

minusminus1

1 23(1) (01) (09) 3(1)(81) 2431(3 1)f = = =

minus

LetLet p p = 10= 10 nn = 3= 3 x x = 1= 1




Tree Diagram



Leaves

(1)

Leaves

(1)

Stays(9)

Stays(9)

33

22

00

22

22



S (9)S (9)

Stays (9)Stays (9)

Stays (9)Stays (9)

S (9)S (9)

S (9)S (9)

S (9)S (9)

L (1)L (1)

L (1)L (1)

L (1)L (1)

L (1)L (1) 00100010

00900090

00900090

72907290

00900090

11

11

08100810

08100810

08100810

1111





n x 05 10 15 20 25 30 35 40 45 50

3 0 8574 7290 6141 5120 4219 3430 2746 2160 1664 1250

1 1354 2430 3251 3840 4219 4410 4436 4320 4084 3750

2 0071 0270 0574 0960 1406 1890 2389 2880 3341 3750

3 0001 0010 0034 0080 0156 0270 0429 0640 0911 1250

p






(1 )np pσ = minus


Var(Var( x x ) =) = σ σ 22



s ExpectedExpected

ValueValue

s VariancVarianc

ee

s StandardStandard

DeviationDeviation









s VarianceVariance




Applications







gets an offer





Answer

Selected = 03456+ 02592+ 00778

=06826

Not Selected = 1-06826 = 03174



Binomial Tables





Applications



















































f xe

x

x

( )

=

minus micro micro

wherewhere



ee = 271828= 271828








weekendweekend

eveningsevenings








4 33 (271828)(4) 1680

4f

minus

= =








000

005

010

015

020

025

0 1 2 3 4 5 6 7 8 9 10


P r o

b a

b i l i t

y



















Example













( )

xe

p x x

λ λ minus

=

45 0

45(0)0

e pminus

=

with λ =45

So p(0) = 001111



Similarly p(1)=004999 and p(2)=011248


is 001111+ 004999 + 011248 =017358















σ = minus(1 )np p








p yp y










Density Function

p y


f x e x( ) =

minus1

micro micro






P x x ex

( )le = minus minus

0 1 o micro

wherewhere









of of


thatthat


22





x x

f ( x )f ( x )

11

33

44

22


Distribution




























6826682668266826









9972997299729972











68266826

95449544

99729972













σ σ = 1= 1

00

z z








z x

=

minus micro

σ









includingincluding


stock of stock of


order isorder is

placedplaced




are beingare being


It haIt ha




1515










= (20 - 15)6= (20 - 15)6= 83= 83


= (20 - 15)6= (20 - 15)6= 83= 83












= 05- 2967= 05- 2967

= 2033= 2033


= 2033= 2033








d d l b bili i ib i












00

Area = 9500Area = 9500

Area = 0500Area = 0500

z z z z

0505








= 15 + 1645(6)= 15 + 1645(6)

= 2487 or 25= 2487 or 25


= 15 + 1645(6)= 15 + 1645(6)

= 2487 or 25= 2487 or 25

























above



Characteristics







the n trials



Binomial Formula


( ) r N r r N r N

r

r N r

r N q pr N r


minus===

)(

)(


000

002

004

006

008

010

012

014

0 5 10 15 20 25 3k

P

( k

5 0

1 3 )

Expectation Value

micro = np = 7 13 = 2333

000

010

020

030

040

0 2 4 6 8 10k

P

( k

7

1 3 )







( )( ) (1 )

( ) x n x n

f x p p x n x

minus= minusminus





Binomial Formulas

Mean

Standard Deviation

( )

( ) ( )

( ) ( ) ( )

Np

p p N p


pm N pq pm

q p p

N

m

m N m N m

N

m

m N m N m

N

m

m N m N m

N

m

m N m N m

N

m

m N m N m

N

m

m N m N m

=




=

sum

minusminusminus

=

minusminus

=

minusminus

=

minusminus

=

minusminus

=

minusminus

=

minus

micro

micro micro

part

part

111

0

1

0

1

0

1

0

1

0

1

0

)1(1)1(

)1()1()1()1(

0)1)((

0

σ 2=


N

sum

P(m N p)m=0

N

sum

=Npq




( )( ) (1 )

( ) x n x n

f x p p x n x

minus= minusminus



FunctionFunction























next yearnext year








f xn

x n x p p x n x( )

( )( )( )

=

minus

minusminus1

1 23(1) (01) (09) 3(1)(81) 2431(3 1)f = = =

minus

LetLet p p = 10= 10 nn = 3= 3 x x = 1= 1




Tree Diagram



Leaves

(1)

Leaves

(1)

Stays(9)

Stays(9)

33

22

00

22

22



S (9)S (9)

Stays (9)Stays (9)

Stays (9)Stays (9)

S (9)S (9)

S (9)S (9)

S (9)S (9)

L (1)L (1)

L (1)L (1)

L (1)L (1)

L (1)L (1) 00100010

00900090

00900090

72907290

00900090

11

11

08100810

08100810

08100810

1111





n x 05 10 15 20 25 30 35 40 45 50

3 0 8574 7290 6141 5120 4219 3430 2746 2160 1664 1250

1 1354 2430 3251 3840 4219 4410 4436 4320 4084 3750

2 0071 0270 0574 0960 1406 1890 2389 2880 3341 3750

3 0001 0010 0034 0080 0156 0270 0429 0640 0911 1250

p






(1 )np pσ = minus


Var(Var( x x ) =) = σ σ 22



s ExpectedExpected

ValueValue

s VariancVarianc

ee

s StandardStandard

DeviationDeviation









s VarianceVariance




Applications







gets an offer





Answer

Selected = 03456+ 02592+ 00778

=06826

Not Selected = 1-06826 = 03174



Binomial Tables





Applications



















































f xe

x

x

( )

=

minus micro micro

wherewhere



ee = 271828= 271828








weekendweekend

eveningsevenings








4 33 (271828)(4) 1680

4f

minus

= =








000

005

010

015

020

025

0 1 2 3 4 5 6 7 8 9 10


P r o

b a

b i l i t

y



















Example













( )

xe

p x x

λ λ minus

=

45 0

45(0)0

e pminus

=

with λ =45

So p(0) = 001111



Similarly p(1)=004999 and p(2)=011248


is 001111+ 004999 + 011248 =017358















σ = minus(1 )np p








p yp y










Density Function

p y


f x e x( ) =

minus1

micro micro






P x x ex

( )le = minus minus

0 1 o micro

wherewhere









of of


thatthat


22





x x

f ( x )f ( x )

11

33

44

22


Distribution





























68266826

95449544

99729972













σ σ = 1= 1

00

z z








z x

=

minus micro

σ









includingincluding


stock of stock of


order isorder is

placedplaced




are beingare being


It haIt ha




1515










= (20 - 15)6= (20 - 15)6= 83= 83


= (20 - 15)6= (20 - 15)6= 83= 83












= 05- 2967= 05- 2967

= 2033= 2033


= 2033= 2033








d d l b bili i ib i












00

Area = 9500Area = 9500

Area = 0500Area = 0500

z z z z

0505








= 15 + 1645(6)= 15 + 1645(6)

= 2487 or 25= 2487 or 25


= 15 + 1645(6)= 15 + 1645(6)

= 2487 or 25= 2487 or 25

























above



Characteristics







the n trials



Binomial Formula


( ) r N r r N r N

r

r N r

r N q pr N r


minus===

)(

)(


000

002

004

006

008

010

012

014

0 5 10 15 20 25 3k

P

( k

5 0

1 3 )

Expectation Value

micro = np = 7 13 = 2333

000

010

020

030

040

0 2 4 6 8 10k

P

( k

7

1 3 )







( )( ) (1 )

( ) x n x n

f x p p x n x

minus= minusminus





Binomial Formulas

Mean

Standard Deviation

( )

( ) ( )

( ) ( ) ( )

Np

p p N p


pm N pq pm

q p p

N

m

m N m N m

N

m

m N m N m

N

m

m N m N m

N

m

m N m N m

N

m

m N m N m

N

m

m N m N m

=




=

sum

minusminusminus

=

minusminus

=

minusminus

=

minusminus

=

minusminus

=

minusminus

=

minus

micro

micro micro

part

part

111

0

1

0

1

0

1

0

1

0

1

0

)1(1)1(

)1()1()1()1(

0)1)((

0

σ 2=


N

sum

P(m N p)m=0

N

sum

=Npq




( )( ) (1 )

( ) x n x n

f x p p x n x

minus= minusminus



FunctionFunction























next yearnext year








f xn

x n x p p x n x( )

( )( )( )

=

minus

minusminus1

1 23(1) (01) (09) 3(1)(81) 2431(3 1)f = = =

minus

LetLet p p = 10= 10 nn = 3= 3 x x = 1= 1




Tree Diagram



Leaves

(1)

Leaves

(1)

Stays(9)

Stays(9)

33

22

00

22

22



S (9)S (9)

Stays (9)Stays (9)

Stays (9)Stays (9)

S (9)S (9)

S (9)S (9)

S (9)S (9)

L (1)L (1)

L (1)L (1)

L (1)L (1)

L (1)L (1) 00100010

00900090

00900090

72907290

00900090

11

11

08100810

08100810

08100810

1111





n x 05 10 15 20 25 30 35 40 45 50

3 0 8574 7290 6141 5120 4219 3430 2746 2160 1664 1250

1 1354 2430 3251 3840 4219 4410 4436 4320 4084 3750

2 0071 0270 0574 0960 1406 1890 2389 2880 3341 3750

3 0001 0010 0034 0080 0156 0270 0429 0640 0911 1250

p






(1 )np pσ = minus


Var(Var( x x ) =) = σ σ 22



s ExpectedExpected

ValueValue

s VariancVarianc

ee

s StandardStandard

DeviationDeviation









s VarianceVariance




Applications







gets an offer





Answer

Selected = 03456+ 02592+ 00778

=06826

Not Selected = 1-06826 = 03174



Binomial Tables





Applications



















































f xe

x

x

( )

=

minus micro micro

wherewhere



ee = 271828= 271828








weekendweekend

eveningsevenings








4 33 (271828)(4) 1680

4f

minus

= =








000

005

010

015

020

025

0 1 2 3 4 5 6 7 8 9 10


P r o

b a

b i l i t

y



















Example













( )

xe

p x x

λ λ minus

=

45 0

45(0)0

e pminus

=

with λ =45

So p(0) = 001111



Similarly p(1)=004999 and p(2)=011248


is 001111+ 004999 + 011248 =017358















σ = minus(1 )np p








p yp y










Density Function

p y


f x e x( ) =

minus1

micro micro






P x x ex

( )le = minus minus

0 1 o micro

wherewhere









of of


thatthat


22





x x

f ( x )f ( x )

11

33

44

22


Distribution































σ σ = 1= 1

00

z z








z x

=

minus micro

σ









includingincluding


stock of stock of


order isorder is

placedplaced




are beingare being


It haIt ha




1515










= (20 - 15)6= (20 - 15)6= 83= 83


= (20 - 15)6= (20 - 15)6= 83= 83












= 05- 2967= 05- 2967

= 2033= 2033


= 2033= 2033








d d l b bili i ib i












00

Area = 9500Area = 9500

Area = 0500Area = 0500

z z z z

0505








= 15 + 1645(6)= 15 + 1645(6)

= 2487 or 25= 2487 or 25


= 15 + 1645(6)= 15 + 1645(6)

= 2487 or 25= 2487 or 25

























above



Characteristics







the n trials



Binomial Formula


( ) r N r r N r N

r

r N r

r N q pr N r


minus===

)(

)(


000

002

004

006

008

010

012

014

0 5 10 15 20 25 3k

P

( k

5 0

1 3 )

Expectation Value

micro = np = 7 13 = 2333

000

010

020

030

040

0 2 4 6 8 10k

P

( k

7

1 3 )







( )( ) (1 )

( ) x n x n

f x p p x n x

minus= minusminus





Binomial Formulas

Mean

Standard Deviation

( )

( ) ( )

( ) ( ) ( )

Np

p p N p


pm N pq pm

q p p

N

m

m N m N m

N

m

m N m N m

N

m

m N m N m

N

m

m N m N m

N

m

m N m N m

N

m

m N m N m

=




=

sum

minusminusminus

=

minusminus

=

minusminus

=

minusminus

=

minusminus

=

minusminus

=

minus

micro

micro micro

part

part

111

0

1

0

1

0

1

0

1

0

1

0

)1(1)1(

)1()1()1()1(

0)1)((

0

σ 2=


N

sum

P(m N p)m=0

N

sum

=Npq




( )( ) (1 )

( ) x n x n

f x p p x n x

minus= minusminus



FunctionFunction























next yearnext year








f xn

x n x p p x n x( )

( )( )( )

=

minus

minusminus1

1 23(1) (01) (09) 3(1)(81) 2431(3 1)f = = =

minus

LetLet p p = 10= 10 nn = 3= 3 x x = 1= 1




Tree Diagram



Leaves

(1)

Leaves

(1)

Stays(9)

Stays(9)

33

22

00

22

22



S (9)S (9)

Stays (9)Stays (9)

Stays (9)Stays (9)

S (9)S (9)

S (9)S (9)

S (9)S (9)

L (1)L (1)

L (1)L (1)

L (1)L (1)

L (1)L (1) 00100010

00900090

00900090

72907290

00900090

11

11

08100810

08100810

08100810

1111





n x 05 10 15 20 25 30 35 40 45 50

3 0 8574 7290 6141 5120 4219 3430 2746 2160 1664 1250

1 1354 2430 3251 3840 4219 4410 4436 4320 4084 3750

2 0071 0270 0574 0960 1406 1890 2389 2880 3341 3750

3 0001 0010 0034 0080 0156 0270 0429 0640 0911 1250

p






(1 )np pσ = minus


Var(Var( x x ) =) = σ σ 22



s ExpectedExpected

ValueValue

s VariancVarianc

ee

s StandardStandard

DeviationDeviation









s VarianceVariance




Applications







gets an offer





Answer

Selected = 03456+ 02592+ 00778

=06826

Not Selected = 1-06826 = 03174



Binomial Tables





Applications



















































f xe

x

x

( )

=

minus micro micro

wherewhere



ee = 271828= 271828








weekendweekend

eveningsevenings








4 33 (271828)(4) 1680

4f

minus

= =








000

005

010

015

020

025

0 1 2 3 4 5 6 7 8 9 10


P r o

b a

b i l i t

y



















Example













( )

xe

p x x

λ λ minus

=

45 0

45(0)0

e pminus

=

with λ =45

So p(0) = 001111



Similarly p(1)=004999 and p(2)=011248


is 001111+ 004999 + 011248 =017358















σ = minus(1 )np p








p yp y










Density Function

p y


f x e x( ) =

minus1

micro micro






P x x ex

( )le = minus minus

0 1 o micro

wherewhere









of of


thatthat


22





x x

f ( x )f ( x )

11

33

44

22


Distribution

























z x

=

minus micro

σ









includingincluding


stock of stock of


order isorder is

placedplaced




are beingare being


It haIt ha




1515










= (20 - 15)6= (20 - 15)6= 83= 83


= (20 - 15)6= (20 - 15)6= 83= 83












= 05- 2967= 05- 2967

= 2033= 2033


= 2033= 2033








d d l b bili i ib i












00

Area = 9500Area = 9500

Area = 0500Area = 0500

z z z z

0505








= 15 + 1645(6)= 15 + 1645(6)

= 2487 or 25= 2487 or 25


= 15 + 1645(6)= 15 + 1645(6)

= 2487 or 25= 2487 or 25

























above



Characteristics







the n trials



Binomial Formula


( ) r N r r N r N

r

r N r

r N q pr N r


minus===

)(

)(


000

002

004

006

008

010

012

014

0 5 10 15 20 25 3k

P

( k

5 0

1 3 )

Expectation Value

micro = np = 7 13 = 2333

000

010

020

030

040

0 2 4 6 8 10k

P

( k

7

1 3 )







( )( ) (1 )

( ) x n x n

f x p p x n x

minus= minusminus





Binomial Formulas

Mean

Standard Deviation

( )

( ) ( )

( ) ( ) ( )

Np

p p N p


pm N pq pm

q p p

N

m

m N m N m

N

m

m N m N m

N

m

m N m N m

N

m

m N m N m

N

m

m N m N m

N

m

m N m N m

=




=

sum

minusminusminus

=

minusminus

=

minusminus

=

minusminus

=

minusminus

=

minusminus

=

minus

micro

micro micro

part

part

111

0

1

0

1

0

1

0

1

0

1

0

)1(1)1(

)1()1()1()1(

0)1)((

0

σ 2=


N

sum

P(m N p)m=0

N

sum

=Npq




( )( ) (1 )

( ) x n x n

f x p p x n x

minus= minusminus



FunctionFunction























next yearnext year








f xn

x n x p p x n x( )

( )( )( )

=

minus

minusminus1

1 23(1) (01) (09) 3(1)(81) 2431(3 1)f = = =

minus

LetLet p p = 10= 10 nn = 3= 3 x x = 1= 1




Tree Diagram



Leaves

(1)

Leaves

(1)

Stays(9)

Stays(9)

33

22

00

22

22



S (9)S (9)

Stays (9)Stays (9)

Stays (9)Stays (9)

S (9)S (9)

S (9)S (9)

S (9)S (9)

L (1)L (1)

L (1)L (1)

L (1)L (1)

L (1)L (1) 00100010

00900090

00900090

72907290

00900090

11

11

08100810

08100810

08100810

1111





n x 05 10 15 20 25 30 35 40 45 50

3 0 8574 7290 6141 5120 4219 3430 2746 2160 1664 1250

1 1354 2430 3251 3840 4219 4410 4436 4320 4084 3750

2 0071 0270 0574 0960 1406 1890 2389 2880 3341 3750

3 0001 0010 0034 0080 0156 0270 0429 0640 0911 1250

p






(1 )np pσ = minus


Var(Var( x x ) =) = σ σ 22



s ExpectedExpected

ValueValue

s VariancVarianc

ee

s StandardStandard

DeviationDeviation









s VarianceVariance




Applications







gets an offer





Answer

Selected = 03456+ 02592+ 00778

=06826

Not Selected = 1-06826 = 03174



Binomial Tables





Applications



















































f xe

x

x

( )

=

minus micro micro

wherewhere



ee = 271828= 271828








weekendweekend

eveningsevenings








4 33 (271828)(4) 1680

4f

minus

= =








000

005

010

015

020

025

0 1 2 3 4 5 6 7 8 9 10


P r o

b a

b i l i t

y



















Example













( )

xe

p x x

λ λ minus

=

45 0

45(0)0

e pminus

=

with λ =45

So p(0) = 001111



Similarly p(1)=004999 and p(2)=011248


is 001111+ 004999 + 011248 =017358















σ = minus(1 )np p








p yp y










Density Function

p y


f x e x( ) =

minus1

micro micro






P x x ex

( )le = minus minus

0 1 o micro

wherewhere









of of


thatthat


22





x x

f ( x )f ( x )

11

33

44

22


Distribution

























includingincluding


stock of stock of


order isorder is

placedplaced




are beingare being


It haIt ha




1515










= (20 - 15)6= (20 - 15)6= 83= 83


= (20 - 15)6= (20 - 15)6= 83= 83












= 05- 2967= 05- 2967

= 2033= 2033


= 2033= 2033








d d l b bili i ib i












00

Area = 9500Area = 9500

Area = 0500Area = 0500

z z z z

0505








= 15 + 1645(6)= 15 + 1645(6)

= 2487 or 25= 2487 or 25


= 15 + 1645(6)= 15 + 1645(6)

= 2487 or 25= 2487 or 25

























above



Characteristics







the n trials



Binomial Formula


( ) r N r r N r N

r

r N r

r N q pr N r


minus===

)(

)(


000

002

004

006

008

010

012

014

0 5 10 15 20 25 3k

P

( k

5 0

1 3 )

Expectation Value

micro = np = 7 13 = 2333

000

010

020

030

040

0 2 4 6 8 10k

P

( k

7

1 3 )







( )( ) (1 )

( ) x n x n

f x p p x n x

minus= minusminus





Binomial Formulas

Mean

Standard Deviation

( )

( ) ( )

( ) ( ) ( )

Np

p p N p


pm N pq pm

q p p

N

m

m N m N m

N

m

m N m N m

N

m

m N m N m

N

m

m N m N m

N

m

m N m N m

N

m

m N m N m

=




=

sum

minusminusminus

=

minusminus

=

minusminus

=

minusminus

=

minusminus

=

minusminus

=

minus

micro

micro micro

part

part

111

0

1

0

1

0

1

0

1

0

1

0

)1(1)1(

)1()1()1()1(

0)1)((

0

σ 2=


N

sum

P(m N p)m=0

N

sum

=Npq




( )( ) (1 )

( ) x n x n

f x p p x n x

minus= minusminus



FunctionFunction























next yearnext year








f xn

x n x p p x n x( )

( )( )( )

=

minus

minusminus1

1 23(1) (01) (09) 3(1)(81) 2431(3 1)f = = =

minus

LetLet p p = 10= 10 nn = 3= 3 x x = 1= 1




Tree Diagram



Leaves

(1)

Leaves

(1)

Stays(9)

Stays(9)

33

22

00

22

22



S (9)S (9)

Stays (9)Stays (9)

Stays (9)Stays (9)

S (9)S (9)

S (9)S (9)

S (9)S (9)

L (1)L (1)

L (1)L (1)

L (1)L (1)

L (1)L (1) 00100010

00900090

00900090

72907290

00900090

11

11

08100810

08100810

08100810

1111





n x 05 10 15 20 25 30 35 40 45 50

3 0 8574 7290 6141 5120 4219 3430 2746 2160 1664 1250

1 1354 2430 3251 3840 4219 4410 4436 4320 4084 3750

2 0071 0270 0574 0960 1406 1890 2389 2880 3341 3750

3 0001 0010 0034 0080 0156 0270 0429 0640 0911 1250

p






(1 )np pσ = minus


Var(Var( x x ) =) = σ σ 22



s ExpectedExpected

ValueValue

s VariancVarianc

ee

s StandardStandard

DeviationDeviation









s VarianceVariance




Applications







gets an offer





Answer

Selected = 03456+ 02592+ 00778

=06826

Not Selected = 1-06826 = 03174



Binomial Tables





Applications



















































f xe

x

x

( )

=

minus micro micro

wherewhere



ee = 271828= 271828








weekendweekend

eveningsevenings








4 33 (271828)(4) 1680

4f

minus

= =








000

005

010

015

020

025

0 1 2 3 4 5 6 7 8 9 10


P r o

b a

b i l i t

y



















Example













( )

xe

p x x

λ λ minus

=

45 0

45(0)0

e pminus

=

with λ =45

So p(0) = 001111



Similarly p(1)=004999 and p(2)=011248


is 001111+ 004999 + 011248 =017358















σ = minus(1 )np p








p yp y










Density Function

p y


f x e x( ) =

minus1

micro micro






P x x ex

( )le = minus minus

0 1 o micro

wherewhere









of of


thatthat


22





x x

f ( x )f ( x )

11

33

44

22


Distribution























are beingare being


It haIt ha




1515










= (20 - 15)6= (20 - 15)6= 83= 83


= (20 - 15)6= (20 - 15)6= 83= 83












= 05- 2967= 05- 2967

= 2033= 2033


= 2033= 2033








d d l b bili i ib i












00

Area = 9500Area = 9500

Area = 0500Area = 0500

z z z z

0505








= 15 + 1645(6)= 15 + 1645(6)

= 2487 or 25= 2487 or 25


= 15 + 1645(6)= 15 + 1645(6)

= 2487 or 25= 2487 or 25

























above



Characteristics







the n trials



Binomial Formula


( ) r N r r N r N

r

r N r

r N q pr N r


minus===

)(

)(


000

002

004

006

008

010

012

014

0 5 10 15 20 25 3k

P

( k

5 0

1 3 )

Expectation Value

micro = np = 7 13 = 2333

000

010

020

030

040

0 2 4 6 8 10k

P

( k

7

1 3 )







( )( ) (1 )

( ) x n x n

f x p p x n x

minus= minusminus





Binomial Formulas

Mean

Standard Deviation

( )

( ) ( )

( ) ( ) ( )

Np

p p N p


pm N pq pm

q p p

N

m

m N m N m

N

m

m N m N m

N

m

m N m N m

N

m

m N m N m

N

m

m N m N m

N

m

m N m N m

=




=

sum

minusminusminus

=

minusminus

=

minusminus

=

minusminus

=

minusminus

=

minusminus

=

minus

micro

micro micro

part

part

111

0

1

0

1

0

1

0

1

0

1

0

)1(1)1(

)1()1()1()1(

0)1)((

0

σ 2=


N

sum

P(m N p)m=0

N

sum

=Npq




( )( ) (1 )

( ) x n x n

f x p p x n x

minus= minusminus



FunctionFunction























next yearnext year








f xn

x n x p p x n x( )

( )( )( )

=

minus

minusminus1

1 23(1) (01) (09) 3(1)(81) 2431(3 1)f = = =

minus

LetLet p p = 10= 10 nn = 3= 3 x x = 1= 1




Tree Diagram



Leaves

(1)

Leaves

(1)

Stays(9)

Stays(9)

33

22

00

22

22



S (9)S (9)

Stays (9)Stays (9)

Stays (9)Stays (9)

S (9)S (9)

S (9)S (9)

S (9)S (9)

L (1)L (1)

L (1)L (1)

L (1)L (1)

L (1)L (1) 00100010

00900090

00900090

72907290

00900090

11

11

08100810

08100810

08100810

1111





n x 05 10 15 20 25 30 35 40 45 50

3 0 8574 7290 6141 5120 4219 3430 2746 2160 1664 1250

1 1354 2430 3251 3840 4219 4410 4436 4320 4084 3750

2 0071 0270 0574 0960 1406 1890 2389 2880 3341 3750

3 0001 0010 0034 0080 0156 0270 0429 0640 0911 1250

p






(1 )np pσ = minus


Var(Var( x x ) =) = σ σ 22



s ExpectedExpected

ValueValue

s VariancVarianc

ee

s StandardStandard

DeviationDeviation









s VarianceVariance




Applications







gets an offer





Answer

Selected = 03456+ 02592+ 00778

=06826

Not Selected = 1-06826 = 03174



Binomial Tables





Applications



















































f xe

x

x

( )

=

minus micro micro

wherewhere



ee = 271828= 271828








weekendweekend

eveningsevenings








4 33 (271828)(4) 1680

4f

minus

= =








000

005

010

015

020

025

0 1 2 3 4 5 6 7 8 9 10


P r o

b a

b i l i t

y



















Example













( )

xe

p x x

λ λ minus

=

45 0

45(0)0

e pminus

=

with λ =45

So p(0) = 001111



Similarly p(1)=004999 and p(2)=011248


is 001111+ 004999 + 011248 =017358















σ = minus(1 )np p








p yp y










Density Function

p y


f x e x( ) =

minus1

micro micro






P x x ex

( )le = minus minus

0 1 o micro

wherewhere









of of


thatthat


22





x x

f ( x )f ( x )

11

33

44

22


Distribution























= (20 - 15)6= (20 - 15)6= 83= 83


= (20 - 15)6= (20 - 15)6= 83= 83












= 05- 2967= 05- 2967

= 2033= 2033


= 2033= 2033








d d l b bili i ib i












00

Area = 9500Area = 9500

Area = 0500Area = 0500

z z z z

0505








= 15 + 1645(6)= 15 + 1645(6)

= 2487 or 25= 2487 or 25


= 15 + 1645(6)= 15 + 1645(6)

= 2487 or 25= 2487 or 25

























above



Characteristics







the n trials



Binomial Formula


( ) r N r r N r N

r

r N r

r N q pr N r


minus===

)(

)(


000

002

004

006

008

010

012

014

0 5 10 15 20 25 3k

P

( k

5 0

1 3 )

Expectation Value

micro = np = 7 13 = 2333

000

010

020

030

040

0 2 4 6 8 10k

P

( k

7

1 3 )







( )( ) (1 )

( ) x n x n

f x p p x n x

minus= minusminus





Binomial Formulas

Mean

Standard Deviation

( )

( ) ( )

( ) ( ) ( )

Np

p p N p


pm N pq pm

q p p

N

m

m N m N m

N

m

m N m N m

N

m

m N m N m

N

m

m N m N m

N

m

m N m N m

N

m

m N m N m

=




=

sum

minusminusminus

=

minusminus

=

minusminus

=

minusminus

=

minusminus

=

minusminus

=

minus

micro

micro micro

part

part

111

0

1

0

1

0

1

0

1

0

1

0

)1(1)1(

)1()1()1()1(

0)1)((

0

σ 2=


N

sum

P(m N p)m=0

N

sum

=Npq




( )( ) (1 )

( ) x n x n

f x p p x n x

minus= minusminus



FunctionFunction























next yearnext year








f xn

x n x p p x n x( )

( )( )( )

=

minus

minusminus1

1 23(1) (01) (09) 3(1)(81) 2431(3 1)f = = =

minus

LetLet p p = 10= 10 nn = 3= 3 x x = 1= 1




Tree Diagram



Leaves

(1)

Leaves

(1)

Stays(9)

Stays(9)

33

22

00

22

22



S (9)S (9)

Stays (9)Stays (9)

Stays (9)Stays (9)

S (9)S (9)

S (9)S (9)

S (9)S (9)

L (1)L (1)

L (1)L (1)

L (1)L (1)

L (1)L (1) 00100010

00900090

00900090

72907290

00900090

11

11

08100810

08100810

08100810

1111





n x 05 10 15 20 25 30 35 40 45 50

3 0 8574 7290 6141 5120 4219 3430 2746 2160 1664 1250

1 1354 2430 3251 3840 4219 4410 4436 4320 4084 3750

2 0071 0270 0574 0960 1406 1890 2389 2880 3341 3750

3 0001 0010 0034 0080 0156 0270 0429 0640 0911 1250

p






(1 )np pσ = minus


Var(Var( x x ) =) = σ σ 22



s ExpectedExpected

ValueValue

s VariancVarianc

ee

s StandardStandard

DeviationDeviation









s VarianceVariance




Applications







gets an offer





Answer

Selected = 03456+ 02592+ 00778

=06826

Not Selected = 1-06826 = 03174



Binomial Tables





Applications



















































f xe

x

x

( )

=

minus micro micro

wherewhere



ee = 271828= 271828








weekendweekend

eveningsevenings








4 33 (271828)(4) 1680

4f

minus

= =








000

005

010

015

020

025

0 1 2 3 4 5 6 7 8 9 10


P r o

b a

b i l i t

y



















Example













( )

xe

p x x

λ λ minus

=

45 0

45(0)0

e pminus

=

with λ =45

So p(0) = 001111



Similarly p(1)=004999 and p(2)=011248


is 001111+ 004999 + 011248 =017358















σ = minus(1 )np p








p yp y










Density Function

p y


f x e x( ) =

minus1

micro micro






P x x ex

( )le = minus minus

0 1 o micro

wherewhere









of of


thatthat


22





x x

f ( x )f ( x )

11

33

44

22


Distribution























= 05- 2967= 05- 2967

= 2033= 2033


= 2033= 2033








d d l b bili i ib i












00

Area = 9500Area = 9500

Area = 0500Area = 0500

z z z z

0505








= 15 + 1645(6)= 15 + 1645(6)

= 2487 or 25= 2487 or 25


= 15 + 1645(6)= 15 + 1645(6)

= 2487 or 25= 2487 or 25

























above



Characteristics







the n trials



Binomial Formula


( ) r N r r N r N

r

r N r

r N q pr N r


minus===

)(

)(


000

002

004

006

008

010

012

014

0 5 10 15 20 25 3k

P

( k

5 0

1 3 )

Expectation Value

micro = np = 7 13 = 2333

000

010

020

030

040

0 2 4 6 8 10k

P

( k

7

1 3 )







( )( ) (1 )

( ) x n x n

f x p p x n x

minus= minusminus





Binomial Formulas

Mean

Standard Deviation

( )

( ) ( )

( ) ( ) ( )

Np

p p N p


pm N pq pm

q p p

N

m

m N m N m

N

m

m N m N m

N

m

m N m N m

N

m

m N m N m

N

m

m N m N m

N

m

m N m N m

=




=

sum

minusminusminus

=

minusminus

=

minusminus

=

minusminus

=

minusminus

=

minusminus

=

minus

micro

micro micro

part

part

111

0

1

0

1

0

1

0

1

0

1

0

)1(1)1(

)1()1()1()1(

0)1)((

0

σ 2=


N

sum

P(m N p)m=0

N

sum

=Npq




( )( ) (1 )

( ) x n x n

f x p p x n x

minus= minusminus



FunctionFunction























next yearnext year








f xn

x n x p p x n x( )

( )( )( )

=

minus

minusminus1

1 23(1) (01) (09) 3(1)(81) 2431(3 1)f = = =

minus

LetLet p p = 10= 10 nn = 3= 3 x x = 1= 1




Tree Diagram



Leaves

(1)

Leaves

(1)

Stays(9)

Stays(9)

33

22

00

22

22



S (9)S (9)

Stays (9)Stays (9)

Stays (9)Stays (9)

S (9)S (9)

S (9)S (9)

S (9)S (9)

L (1)L (1)

L (1)L (1)

L (1)L (1)

L (1)L (1) 00100010

00900090

00900090

72907290

00900090

11

11

08100810

08100810

08100810

1111





n x 05 10 15 20 25 30 35 40 45 50

3 0 8574 7290 6141 5120 4219 3430 2746 2160 1664 1250

1 1354 2430 3251 3840 4219 4410 4436 4320 4084 3750

2 0071 0270 0574 0960 1406 1890 2389 2880 3341 3750

3 0001 0010 0034 0080 0156 0270 0429 0640 0911 1250

p






(1 )np pσ = minus


Var(Var( x x ) =) = σ σ 22



s ExpectedExpected

ValueValue

s VariancVarianc

ee

s StandardStandard

DeviationDeviation









s VarianceVariance




Applications







gets an offer





Answer

Selected = 03456+ 02592+ 00778

=06826

Not Selected = 1-06826 = 03174



Binomial Tables





Applications



















































f xe

x

x

( )

=

minus micro micro

wherewhere



ee = 271828= 271828








weekendweekend

eveningsevenings








4 33 (271828)(4) 1680

4f

minus

= =








000

005

010

015

020

025

0 1 2 3 4 5 6 7 8 9 10


P r o

b a

b i l i t

y



















Example













( )

xe

p x x

λ λ minus

=

45 0

45(0)0

e pminus

=

with λ =45

So p(0) = 001111



Similarly p(1)=004999 and p(2)=011248


is 001111+ 004999 + 011248 =017358















σ = minus(1 )np p








p yp y










Density Function

p y


f x e x( ) =

minus1

micro micro






P x x ex

( )le = minus minus

0 1 o micro

wherewhere









of of


thatthat


22





x x

f ( x )f ( x )

11

33

44

22


Distribution































00

Area = 9500Area = 9500

Area = 0500Area = 0500

z z z z

0505








= 15 + 1645(6)= 15 + 1645(6)

= 2487 or 25= 2487 or 25


= 15 + 1645(6)= 15 + 1645(6)

= 2487 or 25= 2487 or 25

























above



Characteristics







the n trials



Binomial Formula


( ) r N r r N r N

r

r N r

r N q pr N r


minus===

)(

)(


000

002

004

006

008

010

012

014

0 5 10 15 20 25 3k

P

( k

5 0

1 3 )

Expectation Value

micro = np = 7 13 = 2333

000

010

020

030

040

0 2 4 6 8 10k

P

( k

7

1 3 )







( )( ) (1 )

( ) x n x n

f x p p x n x

minus= minusminus





Binomial Formulas

Mean

Standard Deviation

( )

( ) ( )

( ) ( ) ( )

Np

p p N p


pm N pq pm

q p p

N

m

m N m N m

N

m

m N m N m

N

m

m N m N m

N

m

m N m N m

N

m

m N m N m

N

m

m N m N m

=




=

sum

minusminusminus

=

minusminus

=

minusminus

=

minusminus

=

minusminus

=

minusminus

=

minus

micro

micro micro

part

part

111

0

1

0

1

0

1

0

1

0

1

0

)1(1)1(

)1()1()1()1(

0)1)((

0

σ 2=


N

sum

P(m N p)m=0

N

sum

=Npq




( )( ) (1 )

( ) x n x n

f x p p x n x

minus= minusminus



FunctionFunction























next yearnext year








f xn

x n x p p x n x( )

( )( )( )

=

minus

minusminus1

1 23(1) (01) (09) 3(1)(81) 2431(3 1)f = = =

minus

LetLet p p = 10= 10 nn = 3= 3 x x = 1= 1




Tree Diagram



Leaves

(1)

Leaves

(1)

Stays(9)

Stays(9)

33

22

00

22

22



S (9)S (9)

Stays (9)Stays (9)

Stays (9)Stays (9)

S (9)S (9)

S (9)S (9)

S (9)S (9)

L (1)L (1)

L (1)L (1)

L (1)L (1)

L (1)L (1) 00100010

00900090

00900090

72907290

00900090

11

11

08100810

08100810

08100810

1111





n x 05 10 15 20 25 30 35 40 45 50

3 0 8574 7290 6141 5120 4219 3430 2746 2160 1664 1250

1 1354 2430 3251 3840 4219 4410 4436 4320 4084 3750

2 0071 0270 0574 0960 1406 1890 2389 2880 3341 3750

3 0001 0010 0034 0080 0156 0270 0429 0640 0911 1250

p






(1 )np pσ = minus


Var(Var( x x ) =) = σ σ 22



s ExpectedExpected

ValueValue

s VariancVarianc

ee

s StandardStandard

DeviationDeviation









s VarianceVariance




Applications







gets an offer





Answer

Selected = 03456+ 02592+ 00778

=06826

Not Selected = 1-06826 = 03174



Binomial Tables





Applications



















































f xe

x

x

( )

=

minus micro micro

wherewhere



ee = 271828= 271828








weekendweekend

eveningsevenings








4 33 (271828)(4) 1680

4f

minus

= =








000

005

010

015

020

025

0 1 2 3 4 5 6 7 8 9 10


P r o

b a

b i l i t

y



















Example













( )

xe

p x x

λ λ minus

=

45 0

45(0)0

e pminus

=

with λ =45

So p(0) = 001111



Similarly p(1)=004999 and p(2)=011248


is 001111+ 004999 + 011248 =017358















σ = minus(1 )np p








p yp y










Density Function

p y


f x e x( ) =

minus1

micro micro






P x x ex

( )le = minus minus

0 1 o micro

wherewhere









of of


thatthat


22





x x

f ( x )f ( x )

11

33

44

22


Distribution

























= 15 + 1645(6)= 15 + 1645(6)

= 2487 or 25= 2487 or 25


= 15 + 1645(6)= 15 + 1645(6)

= 2487 or 25= 2487 or 25

























above



Characteristics







the n trials



Binomial Formula


( ) r N r r N r N

r

r N r

r N q pr N r


minus===

)(

)(


000

002

004

006

008

010

012

014

0 5 10 15 20 25 3k

P

( k

5 0

1 3 )

Expectation Value

micro = np = 7 13 = 2333

000

010

020

030

040

0 2 4 6 8 10k

P

( k

7

1 3 )







( )( ) (1 )

( ) x n x n

f x p p x n x

minus= minusminus





Binomial Formulas

Mean

Standard Deviation

( )

( ) ( )

( ) ( ) ( )

Np

p p N p


pm N pq pm

q p p

N

m

m N m N m

N

m

m N m N m

N

m

m N m N m

N

m

m N m N m

N

m

m N m N m

N

m

m N m N m

=




=

sum

minusminusminus

=

minusminus

=

minusminus

=

minusminus

=

minusminus

=

minusminus

=

minus

micro

micro micro

part

part

111

0

1

0

1

0

1

0

1

0

1

0

)1(1)1(

)1()1()1()1(

0)1)((

0

σ 2=


N

sum

P(m N p)m=0

N

sum

=Npq




( )( ) (1 )

( ) x n x n

f x p p x n x

minus= minusminus



FunctionFunction























next yearnext year








f xn

x n x p p x n x( )

( )( )( )

=

minus

minusminus1

1 23(1) (01) (09) 3(1)(81) 2431(3 1)f = = =

minus

LetLet p p = 10= 10 nn = 3= 3 x x = 1= 1




Tree Diagram



Leaves

(1)

Leaves

(1)

Stays(9)

Stays(9)

33

22

00

22

22



S (9)S (9)

Stays (9)Stays (9)

Stays (9)Stays (9)

S (9)S (9)

S (9)S (9)

S (9)S (9)

L (1)L (1)

L (1)L (1)

L (1)L (1)

L (1)L (1) 00100010

00900090

00900090

72907290

00900090

11

11

08100810

08100810

08100810

1111





n x 05 10 15 20 25 30 35 40 45 50

3 0 8574 7290 6141 5120 4219 3430 2746 2160 1664 1250

1 1354 2430 3251 3840 4219 4410 4436 4320 4084 3750

2 0071 0270 0574 0960 1406 1890 2389 2880 3341 3750

3 0001 0010 0034 0080 0156 0270 0429 0640 0911 1250

p






(1 )np pσ = minus


Var(Var( x x ) =) = σ σ 22



s ExpectedExpected

ValueValue

s VariancVarianc

ee

s StandardStandard

DeviationDeviation









s VarianceVariance




Applications







gets an offer





Answer

Selected = 03456+ 02592+ 00778

=06826

Not Selected = 1-06826 = 03174



Binomial Tables





Applications



















































f xe

x

x

( )

=

minus micro micro

wherewhere



ee = 271828= 271828








weekendweekend

eveningsevenings








4 33 (271828)(4) 1680

4f

minus

= =








000

005

010

015

020

025

0 1 2 3 4 5 6 7 8 9 10


P r o

b a

b i l i t

y



















Example













( )

xe

p x x

λ λ minus

=

45 0

45(0)0

e pminus

=

with λ =45

So p(0) = 001111



Similarly p(1)=004999 and p(2)=011248


is 001111+ 004999 + 011248 =017358















σ = minus(1 )np p








p yp y










Density Function

p y


f x e x( ) =

minus1

micro micro






P x x ex

( )le = minus minus

0 1 o micro

wherewhere









of of


thatthat


22





x x

f ( x )f ( x )

11

33

44

22


Distribution









































above



Characteristics







the n trials



Binomial Formula


( ) r N r r N r N

r

r N r

r N q pr N r


minus===

)(

)(


000

002

004

006

008

010

012

014

0 5 10 15 20 25 3k

P

( k

5 0

1 3 )

Expectation Value

micro = np = 7 13 = 2333

000

010

020

030

040

0 2 4 6 8 10k

P

( k

7

1 3 )







( )( ) (1 )

( ) x n x n

f x p p x n x

minus= minusminus





Binomial Formulas

Mean

Standard Deviation

( )

( ) ( )

( ) ( ) ( )

Np

p p N p


pm N pq pm

q p p

N

m

m N m N m

N

m

m N m N m

N

m

m N m N m

N

m

m N m N m

N

m

m N m N m

N

m

m N m N m

=




=

sum

minusminusminus

=

minusminus

=

minusminus

=

minusminus

=

minusminus

=

minusminus

=

minus

micro

micro micro

part

part

111

0

1

0

1

0

1

0

1

0

1

0

)1(1)1(

)1()1()1()1(

0)1)((

0

σ 2=


N

sum

P(m N p)m=0

N

sum

=Npq




( )( ) (1 )

( ) x n x n

f x p p x n x

minus= minusminus



FunctionFunction























next yearnext year








f xn

x n x p p x n x( )

( )( )( )

=

minus

minusminus1

1 23(1) (01) (09) 3(1)(81) 2431(3 1)f = = =

minus

LetLet p p = 10= 10 nn = 3= 3 x x = 1= 1




Tree Diagram



Leaves

(1)

Leaves

(1)

Stays(9)

Stays(9)

33

22

00

22

22



S (9)S (9)

Stays (9)Stays (9)

Stays (9)Stays (9)

S (9)S (9)

S (9)S (9)

S (9)S (9)

L (1)L (1)

L (1)L (1)

L (1)L (1)

L (1)L (1) 00100010

00900090

00900090

72907290

00900090

11

11

08100810

08100810

08100810

1111





n x 05 10 15 20 25 30 35 40 45 50

3 0 8574 7290 6141 5120 4219 3430 2746 2160 1664 1250

1 1354 2430 3251 3840 4219 4410 4436 4320 4084 3750

2 0071 0270 0574 0960 1406 1890 2389 2880 3341 3750

3 0001 0010 0034 0080 0156 0270 0429 0640 0911 1250

p






(1 )np pσ = minus


Var(Var( x x ) =) = σ σ 22



s ExpectedExpected

ValueValue

s VariancVarianc

ee

s StandardStandard

DeviationDeviation









s VarianceVariance




Applications







gets an offer





Answer

Selected = 03456+ 02592+ 00778

=06826

Not Selected = 1-06826 = 03174



Binomial Tables





Applications



















































f xe

x

x

( )

=

minus micro micro

wherewhere



ee = 271828= 271828








weekendweekend

eveningsevenings








4 33 (271828)(4) 1680

4f

minus

= =








000

005

010

015

020

025

0 1 2 3 4 5 6 7 8 9 10


P r o

b a

b i l i t

y



















Example













( )

xe

p x x

λ λ minus

=

45 0

45(0)0

e pminus

=

with λ =45

So p(0) = 001111



Similarly p(1)=004999 and p(2)=011248


is 001111+ 004999 + 011248 =017358















σ = minus(1 )np p








p yp y










Density Function

p y


f x e x( ) =

minus1

micro micro






P x x ex

( )le = minus minus

0 1 o micro

wherewhere









of of


thatthat


22





x x

f ( x )f ( x )

11

33

44

22


Distribution






















Characteristics







the n trials



Binomial Formula


( ) r N r r N r N

r

r N r

r N q pr N r


minus===

)(

)(


000

002

004

006

008

010

012

014

0 5 10 15 20 25 3k

P

( k

5 0

1 3 )

Expectation Value

micro = np = 7 13 = 2333

000

010

020

030

040

0 2 4 6 8 10k

P

( k

7

1 3 )







( )( ) (1 )

( ) x n x n

f x p p x n x

minus= minusminus





Binomial Formulas

Mean

Standard Deviation

( )

( ) ( )

( ) ( ) ( )

Np

p p N p


pm N pq pm

q p p

N

m

m N m N m

N

m

m N m N m

N

m

m N m N m

N

m

m N m N m

N

m

m N m N m

N

m

m N m N m

=




=

sum

minusminusminus

=

minusminus

=

minusminus

=

minusminus

=

minusminus

=

minusminus

=

minus

micro

micro micro

part

part

111

0

1

0

1

0

1

0

1

0

1

0

)1(1)1(

)1()1()1()1(

0)1)((

0

σ 2=


N

sum

P(m N p)m=0

N

sum

=Npq




( )( ) (1 )

( ) x n x n

f x p p x n x

minus= minusminus



FunctionFunction























next yearnext year








f xn

x n x p p x n x( )

( )( )( )

=

minus

minusminus1

1 23(1) (01) (09) 3(1)(81) 2431(3 1)f = = =

minus

LetLet p p = 10= 10 nn = 3= 3 x x = 1= 1




Tree Diagram



Leaves

(1)

Leaves

(1)

Stays(9)

Stays(9)

33

22

00

22

22



S (9)S (9)

Stays (9)Stays (9)

Stays (9)Stays (9)

S (9)S (9)

S (9)S (9)

S (9)S (9)

L (1)L (1)

L (1)L (1)

L (1)L (1)

L (1)L (1) 00100010

00900090

00900090

72907290

00900090

11

11

08100810

08100810

08100810

1111





n x 05 10 15 20 25 30 35 40 45 50

3 0 8574 7290 6141 5120 4219 3430 2746 2160 1664 1250

1 1354 2430 3251 3840 4219 4410 4436 4320 4084 3750

2 0071 0270 0574 0960 1406 1890 2389 2880 3341 3750

3 0001 0010 0034 0080 0156 0270 0429 0640 0911 1250

p






(1 )np pσ = minus


Var(Var( x x ) =) = σ σ 22



s ExpectedExpected

ValueValue

s VariancVarianc

ee

s StandardStandard

DeviationDeviation









s VarianceVariance




Applications







gets an offer





Answer

Selected = 03456+ 02592+ 00778

=06826

Not Selected = 1-06826 = 03174



Binomial Tables





Applications



















































f xe

x

x

( )

=

minus micro micro

wherewhere



ee = 271828= 271828








weekendweekend

eveningsevenings








4 33 (271828)(4) 1680

4f

minus

= =








000

005

010

015

020

025

0 1 2 3 4 5 6 7 8 9 10


P r o

b a

b i l i t

y



















Example













( )

xe

p x x

λ λ minus

=

45 0

45(0)0

e pminus

=

with λ =45

So p(0) = 001111



Similarly p(1)=004999 and p(2)=011248


is 001111+ 004999 + 011248 =017358















σ = minus(1 )np p








p yp y










Density Function

p y


f x e x( ) =

minus1

micro micro






P x x ex

( )le = minus minus

0 1 o micro

wherewhere









of of


thatthat


22





x x

f ( x )f ( x )

11

33

44

22


Distribution






















Binomial Formula


( ) r N r r N r N

r

r N r

r N q pr N r


minus===

)(

)(


000

002

004

006

008

010

012

014

0 5 10 15 20 25 3k

P

( k

5 0

1 3 )

Expectation Value

micro = np = 7 13 = 2333

000

010

020

030

040

0 2 4 6 8 10k

P

( k

7

1 3 )







( )( ) (1 )

( ) x n x n

f x p p x n x

minus= minusminus





Binomial Formulas

Mean

Standard Deviation

( )

( ) ( )

( ) ( ) ( )

Np

p p N p


pm N pq pm

q p p

N

m

m N m N m

N

m

m N m N m

N

m

m N m N m

N

m

m N m N m

N

m

m N m N m

N

m

m N m N m

=




=

sum

minusminusminus

=

minusminus

=

minusminus

=

minusminus

=

minusminus

=

minusminus

=

minus

micro

micro micro

part

part

111

0

1

0

1

0

1

0

1

0

1

0

)1(1)1(

)1()1()1()1(

0)1)((

0

σ 2=


N

sum

P(m N p)m=0

N

sum

=Npq




( )( ) (1 )

( ) x n x n

f x p p x n x

minus= minusminus



FunctionFunction























next yearnext year








f xn

x n x p p x n x( )

( )( )( )

=

minus

minusminus1

1 23(1) (01) (09) 3(1)(81) 2431(3 1)f = = =

minus

LetLet p p = 10= 10 nn = 3= 3 x x = 1= 1




Tree Diagram



Leaves

(1)

Leaves

(1)

Stays(9)

Stays(9)

33

22

00

22

22



S (9)S (9)

Stays (9)Stays (9)

Stays (9)Stays (9)

S (9)S (9)

S (9)S (9)

S (9)S (9)

L (1)L (1)

L (1)L (1)

L (1)L (1)

L (1)L (1) 00100010

00900090

00900090

72907290

00900090

11

11

08100810

08100810

08100810

1111





n x 05 10 15 20 25 30 35 40 45 50

3 0 8574 7290 6141 5120 4219 3430 2746 2160 1664 1250

1 1354 2430 3251 3840 4219 4410 4436 4320 4084 3750

2 0071 0270 0574 0960 1406 1890 2389 2880 3341 3750

3 0001 0010 0034 0080 0156 0270 0429 0640 0911 1250

p






(1 )np pσ = minus


Var(Var( x x ) =) = σ σ 22



s ExpectedExpected

ValueValue

s VariancVarianc

ee

s StandardStandard

DeviationDeviation









s VarianceVariance




Applications







gets an offer





Answer

Selected = 03456+ 02592+ 00778

=06826

Not Selected = 1-06826 = 03174



Binomial Tables





Applications



















































f xe

x

x

( )

=

minus micro micro

wherewhere



ee = 271828= 271828








weekendweekend

eveningsevenings








4 33 (271828)(4) 1680

4f

minus

= =








000

005

010

015

020

025

0 1 2 3 4 5 6 7 8 9 10


P r o

b a

b i l i t

y



















Example













( )

xe

p x x

λ λ minus

=

45 0

45(0)0

e pminus

=

with λ =45

So p(0) = 001111



Similarly p(1)=004999 and p(2)=011248


is 001111+ 004999 + 011248 =017358















σ = minus(1 )np p








p yp y










Density Function

p y


f x e x( ) =

minus1

micro micro






P x x ex

( )le = minus minus

0 1 o micro

wherewhere









of of


thatthat


22





x x

f ( x )f ( x )

11

33

44

22


Distribution

























( )( ) (1 )

( ) x n x n

f x p p x n x

minus= minusminus





Binomial Formulas

Mean

Standard Deviation

( )

( ) ( )

( ) ( ) ( )

Np

p p N p


pm N pq pm

q p p

N

m

m N m N m

N

m

m N m N m

N

m

m N m N m

N

m

m N m N m

N

m

m N m N m

N

m

m N m N m

=




=

sum

minusminusminus

=

minusminus

=

minusminus

=

minusminus

=

minusminus

=

minusminus

=

minus

micro

micro micro

part

part

111

0

1

0

1

0

1

0

1

0

1

0

)1(1)1(

)1()1()1()1(

0)1)((

0

σ 2=


N

sum

P(m N p)m=0

N

sum

=Npq




( )( ) (1 )

( ) x n x n

f x p p x n x

minus= minusminus



FunctionFunction























next yearnext year








f xn

x n x p p x n x( )

( )( )( )

=

minus

minusminus1

1 23(1) (01) (09) 3(1)(81) 2431(3 1)f = = =

minus

LetLet p p = 10= 10 nn = 3= 3 x x = 1= 1




Tree Diagram



Leaves

(1)

Leaves

(1)

Stays(9)

Stays(9)

33

22

00

22

22



S (9)S (9)

Stays (9)Stays (9)

Stays (9)Stays (9)

S (9)S (9)

S (9)S (9)

S (9)S (9)

L (1)L (1)

L (1)L (1)

L (1)L (1)

L (1)L (1) 00100010

00900090

00900090

72907290

00900090

11

11

08100810

08100810

08100810

1111





n x 05 10 15 20 25 30 35 40 45 50

3 0 8574 7290 6141 5120 4219 3430 2746 2160 1664 1250

1 1354 2430 3251 3840 4219 4410 4436 4320 4084 3750

2 0071 0270 0574 0960 1406 1890 2389 2880 3341 3750

3 0001 0010 0034 0080 0156 0270 0429 0640 0911 1250

p






(1 )np pσ = minus


Var(Var( x x ) =) = σ σ 22



s ExpectedExpected

ValueValue

s VariancVarianc

ee

s StandardStandard

DeviationDeviation









s VarianceVariance




Applications







gets an offer





Answer

Selected = 03456+ 02592+ 00778

=06826

Not Selected = 1-06826 = 03174



Binomial Tables





Applications



















































f xe

x

x

( )

=

minus micro micro

wherewhere



ee = 271828= 271828








weekendweekend

eveningsevenings








4 33 (271828)(4) 1680

4f

minus

= =








000

005

010

015

020

025

0 1 2 3 4 5 6 7 8 9 10


P r o

b a

b i l i t

y



















Example













( )

xe

p x x

λ λ minus

=

45 0

45(0)0

e pminus

=

with λ =45

So p(0) = 001111



Similarly p(1)=004999 and p(2)=011248


is 001111+ 004999 + 011248 =017358















σ = minus(1 )np p








p yp y










Density Function

p y


f x e x( ) =

minus1

micro micro






P x x ex

( )le = minus minus

0 1 o micro

wherewhere









of of


thatthat


22





x x

f ( x )f ( x )

11

33

44

22


Distribution






















Binomial Formulas

Mean

Standard Deviation

( )

( ) ( )

( ) ( ) ( )

Np

p p N p


pm N pq pm

q p p

N

m

m N m N m

N

m

m N m N m

N

m

m N m N m

N

m

m N m N m

N

m

m N m N m

N

m

m N m N m

=




=

sum

minusminusminus

=

minusminus

=

minusminus

=

minusminus

=

minusminus

=

minusminus

=

minus

micro

micro micro

part

part

111

0

1

0

1

0

1

0

1

0

1

0

)1(1)1(

)1()1()1()1(

0)1)((

0

σ 2=


N

sum

P(m N p)m=0

N

sum

=Npq




( )( ) (1 )

( ) x n x n

f x p p x n x

minus= minusminus



FunctionFunction























next yearnext year








f xn

x n x p p x n x( )

( )( )( )

=

minus

minusminus1

1 23(1) (01) (09) 3(1)(81) 2431(3 1)f = = =

minus

LetLet p p = 10= 10 nn = 3= 3 x x = 1= 1




Tree Diagram



Leaves

(1)

Leaves

(1)

Stays(9)

Stays(9)

33

22

00

22

22



S (9)S (9)

Stays (9)Stays (9)

Stays (9)Stays (9)

S (9)S (9)

S (9)S (9)

S (9)S (9)

L (1)L (1)

L (1)L (1)

L (1)L (1)

L (1)L (1) 00100010

00900090

00900090

72907290

00900090

11

11

08100810

08100810

08100810

1111





n x 05 10 15 20 25 30 35 40 45 50

3 0 8574 7290 6141 5120 4219 3430 2746 2160 1664 1250

1 1354 2430 3251 3840 4219 4410 4436 4320 4084 3750

2 0071 0270 0574 0960 1406 1890 2389 2880 3341 3750

3 0001 0010 0034 0080 0156 0270 0429 0640 0911 1250

p






(1 )np pσ = minus


Var(Var( x x ) =) = σ σ 22



s ExpectedExpected

ValueValue

s VariancVarianc

ee

s StandardStandard

DeviationDeviation









s VarianceVariance




Applications







gets an offer





Answer

Selected = 03456+ 02592+ 00778

=06826

Not Selected = 1-06826 = 03174



Binomial Tables





Applications



















































f xe

x

x

( )

=

minus micro micro

wherewhere



ee = 271828= 271828








weekendweekend

eveningsevenings








4 33 (271828)(4) 1680

4f

minus

= =








000

005

010

015

020

025

0 1 2 3 4 5 6 7 8 9 10


P r o

b a

b i l i t

y



















Example













( )

xe

p x x

λ λ minus

=

45 0

45(0)0

e pminus

=

with λ =45

So p(0) = 001111



Similarly p(1)=004999 and p(2)=011248


is 001111+ 004999 + 011248 =017358















σ = minus(1 )np p








p yp y










Density Function

p y


f x e x( ) =

minus1

micro micro






P x x ex

( )le = minus minus

0 1 o micro

wherewhere









of of


thatthat


22





x x

f ( x )f ( x )

11

33

44

22


Distribution






















( )( ) (1 )

( ) x n x n

f x p p x n x

minus= minusminus



FunctionFunction























next yearnext year








f xn

x n x p p x n x( )

( )( )( )

=

minus

minusminus1

1 23(1) (01) (09) 3(1)(81) 2431(3 1)f = = =

minus

LetLet p p = 10= 10 nn = 3= 3 x x = 1= 1




Tree Diagram



Leaves

(1)

Leaves

(1)

Stays(9)

Stays(9)

33

22

00

22

22



S (9)S (9)

Stays (9)Stays (9)

Stays (9)Stays (9)

S (9)S (9)

S (9)S (9)

S (9)S (9)

L (1)L (1)

L (1)L (1)

L (1)L (1)

L (1)L (1) 00100010

00900090

00900090

72907290

00900090

11

11

08100810

08100810

08100810

1111





n x 05 10 15 20 25 30 35 40 45 50

3 0 8574 7290 6141 5120 4219 3430 2746 2160 1664 1250

1 1354 2430 3251 3840 4219 4410 4436 4320 4084 3750

2 0071 0270 0574 0960 1406 1890 2389 2880 3341 3750

3 0001 0010 0034 0080 0156 0270 0429 0640 0911 1250

p






(1 )np pσ = minus


Var(Var( x x ) =) = σ σ 22



s ExpectedExpected

ValueValue

s VariancVarianc

ee

s StandardStandard

DeviationDeviation









s VarianceVariance




Applications







gets an offer





Answer

Selected = 03456+ 02592+ 00778

=06826

Not Selected = 1-06826 = 03174



Binomial Tables





Applications



















































f xe

x

x

( )

=

minus micro micro

wherewhere



ee = 271828= 271828








weekendweekend

eveningsevenings








4 33 (271828)(4) 1680

4f

minus

= =








000

005

010

015

020

025

0 1 2 3 4 5 6 7 8 9 10


P r o

b a

b i l i t

y



















Example













( )

xe

p x x

λ λ minus

=

45 0

45(0)0

e pminus

=

with λ =45

So p(0) = 001111



Similarly p(1)=004999 and p(2)=011248


is 001111+ 004999 + 011248 =017358















σ = minus(1 )np p








p yp y










Density Function

p y


f x e x( ) =

minus1

micro micro






P x x ex

( )le = minus minus

0 1 o micro

wherewhere









of of


thatthat


22





x x

f ( x )f ( x )

11

33

44

22


Distribution





























next yearnext year








f xn

x n x p p x n x( )

( )( )( )

=

minus

minusminus1

1 23(1) (01) (09) 3(1)(81) 2431(3 1)f = = =

minus

LetLet p p = 10= 10 nn = 3= 3 x x = 1= 1




Tree Diagram



Leaves

(1)

Leaves

(1)

Stays(9)

Stays(9)

33

22

00

22

22



S (9)S (9)

Stays (9)Stays (9)

Stays (9)Stays (9)

S (9)S (9)

S (9)S (9)

S (9)S (9)

L (1)L (1)

L (1)L (1)

L (1)L (1)

L (1)L (1) 00100010

00900090

00900090

72907290

00900090

11

11

08100810

08100810

08100810

1111





n x 05 10 15 20 25 30 35 40 45 50

3 0 8574 7290 6141 5120 4219 3430 2746 2160 1664 1250

1 1354 2430 3251 3840 4219 4410 4436 4320 4084 3750

2 0071 0270 0574 0960 1406 1890 2389 2880 3341 3750

3 0001 0010 0034 0080 0156 0270 0429 0640 0911 1250

p






(1 )np pσ = minus


Var(Var( x x ) =) = σ σ 22



s ExpectedExpected

ValueValue

s VariancVarianc

ee

s StandardStandard

DeviationDeviation









s VarianceVariance




Applications







gets an offer





Answer

Selected = 03456+ 02592+ 00778

=06826

Not Selected = 1-06826 = 03174



Binomial Tables





Applications



















































f xe

x

x

( )

=

minus micro micro

wherewhere



ee = 271828= 271828








weekendweekend

eveningsevenings








4 33 (271828)(4) 1680

4f

minus

= =








000

005

010

015

020

025

0 1 2 3 4 5 6 7 8 9 10


P r o

b a

b i l i t

y



















Example













( )

xe

p x x

λ λ minus

=

45 0

45(0)0

e pminus

=

with λ =45

So p(0) = 001111



Similarly p(1)=004999 and p(2)=011248


is 001111+ 004999 + 011248 =017358















σ = minus(1 )np p








p yp y










Density Function

p y


f x e x( ) =

minus1

micro micro






P x x ex

( )le = minus minus

0 1 o micro

wherewhere









of of


thatthat


22





x x

f ( x )f ( x )

11

33

44

22


Distribution


























f xn

x n x p p x n x( )

( )( )( )

=

minus

minusminus1

1 23(1) (01) (09) 3(1)(81) 2431(3 1)f = = =

minus

LetLet p p = 10= 10 nn = 3= 3 x x = 1= 1




Tree Diagram



Leaves

(1)

Leaves

(1)

Stays(9)

Stays(9)

33

22

00

22

22



S (9)S (9)

Stays (9)Stays (9)

Stays (9)Stays (9)

S (9)S (9)

S (9)S (9)

S (9)S (9)

L (1)L (1)

L (1)L (1)

L (1)L (1)

L (1)L (1) 00100010

00900090

00900090

72907290

00900090

11

11

08100810

08100810

08100810

1111





n x 05 10 15 20 25 30 35 40 45 50

3 0 8574 7290 6141 5120 4219 3430 2746 2160 1664 1250

1 1354 2430 3251 3840 4219 4410 4436 4320 4084 3750

2 0071 0270 0574 0960 1406 1890 2389 2880 3341 3750

3 0001 0010 0034 0080 0156 0270 0429 0640 0911 1250

p






(1 )np pσ = minus


Var(Var( x x ) =) = σ σ 22



s ExpectedExpected

ValueValue

s VariancVarianc

ee

s StandardStandard

DeviationDeviation









s VarianceVariance




Applications







gets an offer





Answer

Selected = 03456+ 02592+ 00778

=06826

Not Selected = 1-06826 = 03174



Binomial Tables





Applications



















































f xe

x

x

( )

=

minus micro micro

wherewhere



ee = 271828= 271828








weekendweekend

eveningsevenings








4 33 (271828)(4) 1680

4f

minus

= =








000

005

010

015

020

025

0 1 2 3 4 5 6 7 8 9 10


P r o

b a

b i l i t

y



















Example













( )

xe

p x x

λ λ minus

=

45 0

45(0)0

e pminus

=

with λ =45

So p(0) = 001111



Similarly p(1)=004999 and p(2)=011248


is 001111+ 004999 + 011248 =017358















σ = minus(1 )np p








p yp y










Density Function

p y


f x e x( ) =

minus1

micro micro






P x x ex

( )le = minus minus

0 1 o micro

wherewhere









of of


thatthat


22





x x

f ( x )f ( x )

11

33

44

22


Distribution






















Tree Diagram



Leaves

(1)

Leaves

(1)

Stays(9)

Stays(9)

33

22

00

22

22



S (9)S (9)

Stays (9)Stays (9)

Stays (9)Stays (9)

S (9)S (9)

S (9)S (9)

S (9)S (9)

L (1)L (1)

L (1)L (1)

L (1)L (1)

L (1)L (1) 00100010

00900090

00900090

72907290

00900090

11

11

08100810

08100810

08100810

1111





n x 05 10 15 20 25 30 35 40 45 50

3 0 8574 7290 6141 5120 4219 3430 2746 2160 1664 1250

1 1354 2430 3251 3840 4219 4410 4436 4320 4084 3750

2 0071 0270 0574 0960 1406 1890 2389 2880 3341 3750

3 0001 0010 0034 0080 0156 0270 0429 0640 0911 1250

p






(1 )np pσ = minus


Var(Var( x x ) =) = σ σ 22



s ExpectedExpected

ValueValue

s VariancVarianc

ee

s StandardStandard

DeviationDeviation









s VarianceVariance




Applications







gets an offer





Answer

Selected = 03456+ 02592+ 00778

=06826

Not Selected = 1-06826 = 03174



Binomial Tables





Applications



















































f xe

x

x

( )

=

minus micro micro

wherewhere



ee = 271828= 271828








weekendweekend

eveningsevenings








4 33 (271828)(4) 1680

4f

minus

= =








000

005

010

015

020

025

0 1 2 3 4 5 6 7 8 9 10


P r o

b a

b i l i t

y



















Example













( )

xe

p x x

λ λ minus

=

45 0

45(0)0

e pminus

=

with λ =45

So p(0) = 001111



Similarly p(1)=004999 and p(2)=011248


is 001111+ 004999 + 011248 =017358















σ = minus(1 )np p








p yp y










Density Function

p y


f x e x( ) =

minus1

micro micro






P x x ex

( )le = minus minus

0 1 o micro

wherewhere









of of


thatthat


22





x x

f ( x )f ( x )

11

33

44

22


Distribution























n x 05 10 15 20 25 30 35 40 45 50

3 0 8574 7290 6141 5120 4219 3430 2746 2160 1664 1250

1 1354 2430 3251 3840 4219 4410 4436 4320 4084 3750

2 0071 0270 0574 0960 1406 1890 2389 2880 3341 3750

3 0001 0010 0034 0080 0156 0270 0429 0640 0911 1250

p






(1 )np pσ = minus


Var(Var( x x ) =) = σ σ 22



s ExpectedExpected

ValueValue

s VariancVarianc

ee

s StandardStandard

DeviationDeviation









s VarianceVariance




Applications







gets an offer





Answer

Selected = 03456+ 02592+ 00778

=06826

Not Selected = 1-06826 = 03174



Binomial Tables





Applications



















































f xe

x

x

( )

=

minus micro micro

wherewhere



ee = 271828= 271828








weekendweekend

eveningsevenings








4 33 (271828)(4) 1680

4f

minus

= =








000

005

010

015

020

025

0 1 2 3 4 5 6 7 8 9 10


P r o

b a

b i l i t

y



















Example













( )

xe

p x x

λ λ minus

=

45 0

45(0)0

e pminus

=

with λ =45

So p(0) = 001111



Similarly p(1)=004999 and p(2)=011248


is 001111+ 004999 + 011248 =017358















σ = minus(1 )np p








p yp y










Density Function

p y


f x e x( ) =

minus1

micro micro






P x x ex

( )le = minus minus

0 1 o micro

wherewhere









of of


thatthat


22





x x

f ( x )f ( x )

11

33

44

22


Distribution























(1 )np pσ = minus


Var(Var( x x ) =) = σ σ 22



s ExpectedExpected

ValueValue

s VariancVarianc

ee

s StandardStandard

DeviationDeviation









s VarianceVariance




Applications







gets an offer





Answer

Selected = 03456+ 02592+ 00778

=06826

Not Selected = 1-06826 = 03174



Binomial Tables





Applications



















































f xe

x

x

( )

=

minus micro micro

wherewhere



ee = 271828= 271828








weekendweekend

eveningsevenings








4 33 (271828)(4) 1680

4f

minus

= =








000

005

010

015

020

025

0 1 2 3 4 5 6 7 8 9 10


P r o

b a

b i l i t

y



















Example













( )

xe

p x x

λ λ minus

=

45 0

45(0)0

e pminus

=

with λ =45

So p(0) = 001111



Similarly p(1)=004999 and p(2)=011248


is 001111+ 004999 + 011248 =017358















σ = minus(1 )np p








p yp y










Density Function

p y


f x e x( ) =

minus1

micro micro






P x x ex

( )le = minus minus

0 1 o micro

wherewhere









of of


thatthat


22





x x

f ( x )f ( x )

11

33

44

22


Distribution



























s VarianceVariance




Applications







gets an offer





Answer

Selected = 03456+ 02592+ 00778

=06826

Not Selected = 1-06826 = 03174



Binomial Tables





Applications



















































f xe

x

x

( )

=

minus micro micro

wherewhere



ee = 271828= 271828








weekendweekend

eveningsevenings








4 33 (271828)(4) 1680

4f

minus

= =








000

005

010

015

020

025

0 1 2 3 4 5 6 7 8 9 10


P r o

b a

b i l i t

y



















Example













( )

xe

p x x

λ λ minus

=

45 0

45(0)0

e pminus

=

with λ =45

So p(0) = 001111



Similarly p(1)=004999 and p(2)=011248


is 001111+ 004999 + 011248 =017358















σ = minus(1 )np p








p yp y










Density Function

p y


f x e x( ) =

minus1

micro micro






P x x ex

( )le = minus minus

0 1 o micro

wherewhere









of of


thatthat


22





x x

f ( x )f ( x )

11

33

44

22


Distribution






















Applications







gets an offer





Answer

Selected = 03456+ 02592+ 00778

=06826

Not Selected = 1-06826 = 03174



Binomial Tables





Applications



















































f xe

x

x

( )

=

minus micro micro

wherewhere



ee = 271828= 271828








weekendweekend

eveningsevenings








4 33 (271828)(4) 1680

4f

minus

= =








000

005

010

015

020

025

0 1 2 3 4 5 6 7 8 9 10


P r o

b a

b i l i t

y



















Example













( )

xe

p x x

λ λ minus

=

45 0

45(0)0

e pminus

=

with λ =45

So p(0) = 001111



Similarly p(1)=004999 and p(2)=011248


is 001111+ 004999 + 011248 =017358















σ = minus(1 )np p








p yp y










Density Function

p y


f x e x( ) =

minus1

micro micro






P x x ex

( )le = minus minus

0 1 o micro

wherewhere









of of


thatthat


22





x x

f ( x )f ( x )

11

33

44

22


Distribution






















Answer

Selected = 03456+ 02592+ 00778

=06826

Not Selected = 1-06826 = 03174



Binomial Tables





Applications



















































f xe

x

x

( )

=

minus micro micro

wherewhere



ee = 271828= 271828








weekendweekend

eveningsevenings








4 33 (271828)(4) 1680

4f

minus

= =








000

005

010

015

020

025

0 1 2 3 4 5 6 7 8 9 10


P r o

b a

b i l i t

y



















Example













( )

xe

p x x

λ λ minus

=

45 0

45(0)0

e pminus

=

with λ =45

So p(0) = 001111



Similarly p(1)=004999 and p(2)=011248


is 001111+ 004999 + 011248 =017358















σ = minus(1 )np p








p yp y










Density Function

p y


f x e x( ) =

minus1

micro micro






P x x ex

( )le = minus minus

0 1 o micro

wherewhere









of of


thatthat


22





x x

f ( x )f ( x )

11

33

44

22


Distribution






















Binomial Tables





Applications



















































f xe

x

x

( )

=

minus micro micro

wherewhere



ee = 271828= 271828








weekendweekend

eveningsevenings








4 33 (271828)(4) 1680

4f

minus

= =








000

005

010

015

020

025

0 1 2 3 4 5 6 7 8 9 10


P r o

b a

b i l i t

y



















Example













( )

xe

p x x

λ λ minus

=

45 0

45(0)0

e pminus

=

with λ =45

So p(0) = 001111



Similarly p(1)=004999 and p(2)=011248


is 001111+ 004999 + 011248 =017358















σ = minus(1 )np p








p yp y










Density Function

p y


f x e x( ) =

minus1

micro micro






P x x ex

( )le = minus minus

0 1 o micro

wherewhere









of of


thatthat


22





x x

f ( x )f ( x )

11

33

44

22


Distribution






















Applications



















































f xe

x

x

( )

=

minus micro micro

wherewhere



ee = 271828= 271828








weekendweekend

eveningsevenings








4 33 (271828)(4) 1680

4f

minus

= =








000

005

010

015

020

025

0 1 2 3 4 5 6 7 8 9 10


P r o

b a

b i l i t

y



















Example













( )

xe

p x x

λ λ minus

=

45 0

45(0)0

e pminus

=

with λ =45

So p(0) = 001111



Similarly p(1)=004999 and p(2)=011248


is 001111+ 004999 + 011248 =017358















σ = minus(1 )np p








p yp y










Density Function

p y


f x e x( ) =

minus1

micro micro






P x x ex

( )le = minus minus

0 1 o micro

wherewhere









of of


thatthat


22





x x

f ( x )f ( x )

11

33

44

22


Distribution





























































f xe

x

x

( )

=

minus micro micro

wherewhere



ee = 271828= 271828








weekendweekend

eveningsevenings








4 33 (271828)(4) 1680

4f

minus

= =








000

005

010

015

020

025

0 1 2 3 4 5 6 7 8 9 10


P r o

b a

b i l i t

y



















Example













( )

xe

p x x

λ λ minus

=

45 0

45(0)0

e pminus

=

with λ =45

So p(0) = 001111



Similarly p(1)=004999 and p(2)=011248


is 001111+ 004999 + 011248 =017358















σ = minus(1 )np p








p yp y










Density Function

p y


f x e x( ) =

minus1

micro micro






P x x ex

( )le = minus minus

0 1 o micro

wherewhere









of of


thatthat


22





x x

f ( x )f ( x )

11

33

44

22


Distribution


























weekendweekend

eveningsevenings








4 33 (271828)(4) 1680

4f

minus

= =








000

005

010

015

020

025

0 1 2 3 4 5 6 7 8 9 10


P r o

b a

b i l i t

y



















Example













( )

xe

p x x

λ λ minus

=

45 0

45(0)0

e pminus

=

with λ =45

So p(0) = 001111



Similarly p(1)=004999 and p(2)=011248


is 001111+ 004999 + 011248 =017358















σ = minus(1 )np p








p yp y










Density Function

p y


f x e x( ) =

minus1

micro micro






P x x ex

( )le = minus minus

0 1 o micro

wherewhere









of of


thatthat


22





x x

f ( x )f ( x )

11

33

44

22


Distribution
























4 33 (271828)(4) 1680

4f

minus

= =








000

005

010

015

020

025

0 1 2 3 4 5 6 7 8 9 10


P r o

b a

b i l i t

y



















Example













( )

xe

p x x

λ λ minus

=

45 0

45(0)0

e pminus

=

with λ =45

So p(0) = 001111



Similarly p(1)=004999 and p(2)=011248


is 001111+ 004999 + 011248 =017358















σ = minus(1 )np p








p yp y










Density Function

p y


f x e x( ) =

minus1

micro micro






P x x ex

( )le = minus minus

0 1 o micro

wherewhere









of of


thatthat


22





x x

f ( x )f ( x )

11

33

44

22


Distribution

























000

005

010

015

020

025

0 1 2 3 4 5 6 7 8 9 10


P r o

b a

b i l i t

y



















Example













( )

xe

p x x

λ λ minus

=

45 0

45(0)0

e pminus

=

with λ =45

So p(0) = 001111



Similarly p(1)=004999 and p(2)=011248


is 001111+ 004999 + 011248 =017358















σ = minus(1 )np p








p yp y










Density Function

p y


f x e x( ) =

minus1

micro micro






P x x ex

( )le = minus minus

0 1 o micro

wherewhere









of of


thatthat


22





x x

f ( x )f ( x )

11

33

44

22


Distribution



































Example













( )

xe

p x x

λ λ minus

=

45 0

45(0)0

e pminus

=

with λ =45

So p(0) = 001111



Similarly p(1)=004999 and p(2)=011248


is 001111+ 004999 + 011248 =017358















σ = minus(1 )np p








p yp y










Density Function

p y


f x e x( ) =

minus1

micro micro






P x x ex

( )le = minus minus

0 1 o micro

wherewhere









of of


thatthat


22





x x

f ( x )f ( x )

11

33

44

22


Distribution
























( )

xe

p x x

λ λ minus

=

45 0

45(0)0

e pminus

=

with λ =45

So p(0) = 001111



Similarly p(1)=004999 and p(2)=011248


is 001111+ 004999 + 011248 =017358















σ = minus(1 )np p








p yp y










Density Function

p y


f x e x( ) =

minus1

micro micro






P x x ex

( )le = minus minus

0 1 o micro

wherewhere









of of


thatthat


22





x x

f ( x )f ( x )

11

33

44

22


Distribution






















Similarly p(1)=004999 and p(2)=011248


is 001111+ 004999 + 011248 =017358















σ = minus(1 )np p








p yp y










Density Function

p y


f x e x( ) =

minus1

micro micro






P x x ex

( )le = minus minus

0 1 o micro

wherewhere









of of


thatthat


22





x x

f ( x )f ( x )

11

33

44

22


Distribution

































σ = minus(1 )np p








p yp y










Density Function

p y


f x e x( ) =

minus1

micro micro






P x x ex

( )le = minus minus

0 1 o micro

wherewhere









of of


thatthat


22





x x

f ( x )f ( x )

11

33

44

22


Distribution






















p yp y










Density Function

p y


f x e x( ) =

minus1

micro micro






P x x ex

( )le = minus minus

0 1 o micro

wherewhere









of of


thatthat


22





x x

f ( x )f ( x )

11

33

44

22


Distribution






















Density Function

p y


f x e x( ) =

minus1

micro micro






P x x ex

( )le = minus minus

0 1 o micro

wherewhere









of of


thatthat


22





x x

f ( x )f ( x )

11

33

44

22


Distribution
























P x x ex

( )le = minus minus

0 1 o micro

wherewhere









of of


thatthat


22





x x

f ( x )f ( x )

11

33

44

22


Distribution


























of of


thatthat


22





x x

f ( x )f ( x )

11

33

44

22


Distribution






















x x

f ( x )f ( x )

11

33

44

22


Distribution




















Normal +Bionomial

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Transcript of Normal +Bionomial