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Chapter 3Discrete Random Variables and Probability Distributions

Presenting the Theoretical Distributions Uniform Binomial Geometric Poisson

Chapter 3BENM 500 campusstudents excited about today’s lecture

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What You Should Know about the Upcoming Discrete Distributions

The type of stochastic situations that give rise to the distribution.

What is the ‘classic’ scenario? What distinguishes this distribution’s application

from the others? The type of real world situations you can

model with these distributions. How do you turn the crank and generate

some valid probabilities given a scenario description?

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Discrete Uniform DistributionDefinition: a random variable X is a discrete

uniform random variable if each of the n values in its range {x1, x2,….xn} has equal probability. Then f(xi) = 1/n

“Discrete” implies that only specific values within the range are possible.

For example, a digital weight scale reads between 0 and 10 pounds and has two decimal place accuracy.

The possible values are {0.00, 0.01, 0.02, … , 10.00}. Other values, such as 7.6382 are in the range but are not possible values.

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3-5 Discrete Uniform Distribution

Definition

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3-5 Discrete Uniform Distribution

Example 3-13

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3-5 Discrete Uniform Distribution

Figure 3-7 Probability mass function for a discrete uniform random variable.

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3-5 Discrete Uniform Distribution

Mean and Variance

( , )X U a b“follows”

1( )

1f x

b a

parameters

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Finding the Mean

1[ ]

1 1

1 12 ( )

1 2 2

b b

x a x a

xE X x

b a b a

b a b aa b a

b a

12 ( 1)

2nS n a n d

The sum of an arithmetic series where a is the first term, n is the number of terms, and d= is the difference between consecutive values.

( , )X U a b

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A Uniform Example The number of demands per day for a symmetrical,

spiral, closed-ended sprocket is

(0,24)X U

2

0 241( ) ; [ ] 12

25 2

(24 1) 1( ) 52; 7.211

121

( )25

11 14Pr{ 10} 1 (10) 1 .56

25 25

f x E X

Var X

xF x

X F

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3-6 Binomial Distribution

Random experiments and random variables

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3-6 Binomial Distribution

Random experiments and random variables

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Binomial Distribution

Many random experiments produce results that fall into patterns. One such pattern can be characterized as follows:

A series of independent random trials occurs Each trial can be summarized as a “success” or a

“failure” – called a Bernoulli trial The probability of success on each trial (p) remains

constant The random variable of interest is a count of the

number of successes in n trials.

IndependentBernoulli trials

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Jacob Bernoulli

Born December 27, 1654)Basel, Switzerland

Died August 16, 1705 (aged 50)Basel, Switzerland

Nationality    Swiss

Field Mathematician

Institutions University of Basel

Academic advisor   Gottfried Leibniz

Notable students   Johann BernoulliJacob HermannNicolaus I Bernoulli

Known for Bernoulli trialBernoulli numbers

Religion Calvinist

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3-6 Binomial Distribution

Definition( , )X B n p

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Deriving the PMF

!

( ) (1 ) (1 )! !

x n x x n xn nf x p p p p

x x n x

Let n = 7 and p = .2

3 7 37!

(3) .2 (1 .2)3! 7 3 !

f

X = success0 = failure

X 0 0 X 0 X 0

This is just the number of permutations of 7 objects where 3 are the same and the other 4 are the same.

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Is this a PMF?

0

(1 ) (1 ) 1n

nx n x

x

np p p p

x

0

( )n

n n r r

r

na b a b

r

Binomial Theorem:

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3-6 Binomial Distribution

Figure 3-8 Binomial distributions for selected values of n and p.

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3-6 Binomial Distribution

Example 3-18

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3-6 Binomial Distribution

Example 3-18

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Mean & Variance of a Binomial Distribution

Let X = number of successes in n trials

1 if kth trial is a success

0 otherwisekX

1

n

kk

X X

( ) ( ) (1) (0)(1 )kx

E X x f x p p p

2 2 2( ) ( ) ( ) (1 ) (0 ) (1 )kx

V X x f x p p p p (1 )p p

1 1

( ) ( )n n

kk k

E X E X p np

2

1 1

( ) ( ) (1 ) (1 )n n

kk k

V X V X p p np p

What value of p gives the largest

variance or uncertainty?

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3-6 Binomial Distribution

Mean and Variance ( , )X B n p

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Problem 3-79 (overbooking)Assume 120 seats are available on an airliner, 125

tickets were sold. The probability a passenger does not show is 0.10. X = actual number of no-shows.

What is the probability that every passenger who shows up can take this flight?

What is the probability that the flight departs with empty seats?

(125,.1), [ ] 12.5, [ ] 11.25X B E X V X

Pr{ 5} 1 (4) 1 .003859 .9961X F

Pr{ 6} 1 (5) 1 .011432 .988568X F

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A Binomial Example The probability of a forty-year old male dying in his

40th year is .002589. A group of 20 forty-year old males meet at their high school reunion. What is the probability that at least one of them will be dead before the year is over?

Let X = a discrete random variable, the number of deathsamong 20 40-year old males. X = 0, 1, 2, …, 20.

(20,.002589), [ ] .05178, [ ] .05765X B E X V X

Pr{X 1} = 1 – f(0) = 1 - .9495 = .0505

20(0) (1 .002589)f

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3-7 Geometric Distributions

Definition ( )X Geo p

sample spaceS={s, fs, ffs, fffs, …}

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3-7 Geometric Distributions

Figure 3-9. Geometric distributions for selected values of the parameter p.

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3-7 Geometric Distributions

3-7.1 Geometric Distribution

Example 3-21

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3-7 Geometric Distributions ( )X Geo p

1

1 1

2

2 2 2 12

1 1

1[ ] ( ) (1 ) " "

1 1[ ] ( ) (1 ) " "

x

x x

x

x x

E X xp x x p p gapp

pV X x f x x p p gap

p p

The overachieving studentmay wish to fill in the gaps

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Geometric Distribution

1

1 1

1 1 1

1

1 1

( ) (1 ) ; 1, 2,3,...

( ) (1 ) (1 ) 11 (1 )

1 1( ) ( ) (1 ) 1 1

x

x x

x x x

xx x

xi

i i

f x p p x

pf x p p p p

p

p pF x p i p p p

p

1

1

n

n

a rS

r

lim , 11n

n

aS if r

r

Geometric seriesp = ar = 1 -p

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A Geometric example

There is one chance in 10,000 of winning a particular lottery game. How many games must be played to achieve the first win?

Let X = a discrete random variable, the number of gamesplayed to achieve the first win. (.0001)X Geo

1

365

( ) .0001) .9999

1[ ] 10,000; [ ] 99,990,000; 9999.5

.0001

( ) 1 .9999

Pr{ 365} 1 .9999 .0358

x

x

f x

E X Var X

F x

X

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3-7 Geometric Distributions

Lack of Memory Property

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3-9 Poisson Distribution

Definition

( )X Pois

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Situations That Lead to the Poisson

Telephone calls arriving at a help desk Number of alpha particles emitted from a

radioactive source Number of accidents occurring over a given time

period (used by insurance industry) Other examples are customers arriving at a store,

bank, or fast food outlet. In light traffic, the number of vehicles that pass a

marker on a roadway The arrival of natural events such as tornadoes,

hurricanes, and lightning strikes.

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3-9 Poisson Distribution

Mean and Variance

1 1

0 0 1 1

( )! ( 1)! ( 1)!

x x x

x x x x

e x xxp x e e e e

x x x x

( )X Pois

2 3

0

1 ...2! 3! !

nx

n

x x xe x

n

The overachievingstudent may wishto derive the variance

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3-9 Poisson Distribution

Consistent Units

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3-9 Poisson Distribution

Example 3-33

( )!

xep x

x

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3-9 Poisson Distribution

Example 3-33

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Problem 3-110

Telephone calls arrive at a phone exchange with Poisson Distribution; = 10 calls / hour

What is the probability there are 3 or fewer calls in one hour?

What is the probability that there are exactly 15 calls in two hours?

10 5(10)

( 5) 0.0378! 5!

xe eP X

x

20 15(20)( 15) 0.05165

! 15!

xe eP X

x

103

0

(10)( 3) .01034

!

x

x

eP X

x

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Another Poisson Problem

The number of automobile accidents on I-75 passing through downtown Dayton has a Poisson distribution with a mean of 3 per week. What is the probability of at least one accident a week?

3 0(3)( 1) 1 ( 0} 1 .9502

0!

eP X P X

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Summary of DistributionsDistribution Mean Variance Variance / Mean

Uniform(a+b)/2

[(b-a+1)2 – 1] / 12[(b-a+1)2 – 1] / [2(b+a)]

Binomial np np(1 - p) (1 – p) < 1

Geometric 1/p (1 – p)/p2 (1 – p) / p

Neg. Binomial

r/p r(1-p)/p2 (1 - p) / p

Hypergeom. np np(1-p)(N-n)/(N-1) (1-p)(N-n)/(N-1)

Poisson 1

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Other Discrete Distributions Worth Knowing

Negative Binomial a generalization of the geometric distribution number of Bernoulli trials until the rth success

Hypergeometric used to model finite populations parameters are

N – population size n – sample size K – number of successes in population

X = a discrete random variable, the number of “successes” in the sample

probabilities computed from combinations

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Next Week…Chapter 4