Chapter 5 Discrete Probability Distributions

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© 2006 Thomson/South-Western© 2006 Thomson/South-Western

Chapter 5Chapter 5 Discrete Probability Distributions Discrete Probability Distributions

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Random VariablesRandom Variables Discrete Probability DistributionsDiscrete Probability Distributions Expected Value and VarianceExpected Value and Variance Binomial DistributionBinomial Distribution

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A A random variablerandom variable is a numerical description of the is a numerical description of the outcome of an experiment.outcome of an experiment. A A random variablerandom variable is a numerical description of the is a numerical description of the outcome of an experiment.outcome of an experiment.

Random VariablesRandom Variables

A A discrete random variablediscrete random variable may assume either a may assume either a finite number of values or an infinite sequence offinite number of values or an infinite sequence of values.values.

A A discrete random variablediscrete random variable may assume either a may assume either a finite number of values or an infinite sequence offinite number of values or an infinite sequence of values.values.

A A continuous random variablecontinuous random variable may assume any may assume any numerical value in an interval or collection ofnumerical value in an interval or collection of intervals.intervals.

A A continuous random variablecontinuous random variable may assume any may assume any numerical value in an interval or collection ofnumerical value in an interval or collection of intervals.intervals.

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Let Let xx = number of TVs sold at the store in one day, = number of TVs sold at the store in one day,

where where xx can take on 5 values (0, 1, 2, 3, 4) can take on 5 values (0, 1, 2, 3, 4)

Let Let xx = number of TVs sold at the store in one day, = number of TVs sold at the store in one day,

where where xx can take on 5 values (0, 1, 2, 3, 4) can take on 5 values (0, 1, 2, 3, 4)

Example: JSL AppliancesExample: JSL Appliances

Discrete random variable with a Discrete random variable with a finitefinite number number of values of values

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Let Let xx = number of customers arriving in one day, = number of customers arriving in one day,

where where xx can take on the values 0, 1, 2, . . . can take on the values 0, 1, 2, . . .

Let Let xx = number of customers arriving in one day, = number of customers arriving in one day,

where where xx can take on the values 0, 1, 2, . . . can take on the values 0, 1, 2, . . .

Example: JSL AppliancesExample: JSL Appliances

Discrete random variable with an Discrete 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 nofinite upper limit on the number that might arrive.finite upper limit on the number that might arrive.

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Random VariablesRandom Variables

QuestionQuestion Random Variable Random Variable xx TypeType

FamilyFamilysizesize

xx = Number of dependents = Number of dependents reported on tax returnreported on tax return

DiscreteDiscrete

Distance fromDistance fromhome to storehome to store

xx = Distance in miles from = Distance in miles from home to the store sitehome to the store site

ContinuousContinuous

Own dogOwn dogor cator cat

xx = 1 if own no pet; = 1 if own no pet; = 2 if own dog(s) only; = 2 if own dog(s) only; = 3 if own cat(s) only; = 3 if own cat(s) only; = 4 if own dog(s) and cat(s)= 4 if own dog(s) and cat(s)

DiscreteDiscrete

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The The probability distributionprobability distribution for a random variable for a random variable describes how probabilities are distributed overdescribes how probabilities are distributed over the values of the random variable.the values of the random variable.

The The probability distributionprobability distribution for a random variable for a random variable describes how probabilities are distributed overdescribes how probabilities are distributed over the values of the random variable.the values of the random variable.

We can describe a discrete probability distributionWe can describe a discrete probability distribution with a table, graph, or equation.with a table, graph, or equation. We can describe a discrete probability distributionWe can describe a discrete probability distribution with a table, graph, or equation.with a table, graph, or equation.

Discrete Probability DistributionsDiscrete Probability Distributions

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The probability distribution is defined by aThe probability distribution is defined by a probability functionprobability function, denoted by , denoted by ff((xx), which provides), which provides the probability for each value of the random variable.the probability for each value of the random variable.

The probability distribution is defined by aThe probability distribution is defined by a probability functionprobability function, denoted by , denoted by ff((xx), which provides), which provides the probability for each value of the random variable.the probability for each value of the random variable.

The required conditions for a discrete probabilityThe required conditions for a discrete probability function are:function are: The required conditions for a discrete probabilityThe required conditions for a discrete probability function are:function are:


ff((xx) ) >> 0 0ff((xx) ) >> 0 0

ff((xx) = 1) = 1ff((xx) = 1) = 1

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a tabular representation of the probabilitya tabular representation of the probability distribution for TV sales was developed.distribution for TV sales was developed.

Using past data on TV sales, …Using past data on TV sales, …

NumberNumber Units SoldUnits Sold of Daysof Days

00 80 80 11 50 50 22 40 40 33 10 10 44 20 20

200200

xx ff((xx)) 00 .40 .40 11 .25 .25 22 .20 .20 33 .05 .05 44 .10 .10

1.001.00

80/20080/200


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0 1 2 3 40 1 2 3 4Values of Random Variable Values of Random Variable xx (TV sales) (TV sales)Values of Random Variable Values of Random Variable xx (TV sales) (TV sales)

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babili

tyPro

babili

tyPro

babili

tyPro

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ty


Graphical Representation of Probability DistributionGraphical Representation of Probability Distribution

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Expected Value and VarianceExpected Value and Variance

The The expected valueexpected value, or mean, of a random variable, or mean, of a random variable is a measure of its central location.is a measure of its central location. The The expected valueexpected value, or mean, of a random variable, or mean, of a random variable is a measure of its central location.is a measure of its central location.

The The variancevariance summarizes the variability in the summarizes the variability in the values of a random variable.values of a random variable. The The variancevariance summarizes the variability in the summarizes the variability in the values of a random variable.values of a random variable.

The The standard deviationstandard deviation, , , is defined as the positive, is defined as the positive square root of the variance.square root of the variance. The The standard deviationstandard deviation, , , is defined as the positive, is defined as the positive square root of the variance.square root of the variance.

Var(Var(xx) = ) = 22 = = ((xx - - ))22ff((xx))Var(Var(xx) = ) = 22 = = ((xx - - ))22ff((xx))

EE((xx) = ) = = = xfxf((xx))EE((xx) = ) = = = xfxf((xx))

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Expected ValueExpected Value

expected number expected number of TVs sold in a dayof TVs sold in a day

xx ff((xx)) xfxf((xx))

00 .40 .40 .00 .00

11 .25 .25 .25 .25

22 .20 .20 .40 .40

33 .05 .05 .15 .15

44 .10 .10 .40.40

EE((xx) = 1.20) = 1.20


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Variance and Standard DeviationVariance and Standard Deviation

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11

22

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44

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-0.2-0.2

0.80.8

1.81.8

2.82.8

1.441.44

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0.640.64

3.243.24

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x - x - ((x - x - ))22 ff((xx)) ((xx - - ))22ff((xx))

Variance of daily sales = Variance of daily sales = 22 = 1.660 = 1.660

xx

TVsTVssquaresquare

dd

Standard deviation of daily sales = 1.2884 TVsStandard deviation of daily sales = 1.2884 TVs


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Binomial DistributionBinomial Distribution

Four Properties of a Binomial ExperimentFour Properties of a Binomial Experiment

3. The probability of a success, denoted by 3. The probability of a success, denoted by pp, does, does not change from trial to trial.not change from trial to trial.3. The probability of a success, denoted by 3. The probability of a success, denoted by pp, does, does not change from trial to trial.not change from trial to trial.

4. The trials are independent.4. The trials are independent.4. The trials are independent.4. The trials are independent.

2. Two outcomes, 2. Two outcomes, successsuccess and and failurefailure, are possible, are possible on each trial.on each trial.2. Two outcomes, 2. Two outcomes, successsuccess and and failurefailure, are possible, are possible on each trial.on each trial.

1. The experiment consists of a sequence of 1. The experiment consists of a sequence of nn identical trials.identical trials.1. The experiment consists of a sequence of 1. The experiment consists of a sequence of nn identical trials.identical trials.

stationaritstationarityy

assumptioassumptionn

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Our interest is in the Our interest is in the number of successesnumber of successes occurring in the occurring in the nn trials. trials. Our interest is in the Our interest is in the number of successesnumber of successes occurring in the occurring in the nn trials. trials.

We let We let xx denote the number of successes denote the number of successes occurring in the occurring in the nn trials. trials. We let We let xx denote the number of successes denote the number of successes occurring in the occurring in the nn trials. trials.

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where:where: ff((xx) = the probability of ) = the probability of xx successes in successes in nn trials trials nn = the number of trials = the number of trials pp = the probability of success on any one trial = the probability of success on any one trial

( )!( ) (1 )

!( )!x n xn

f x p px n x

( )!( ) (1 )

!( )!x n xn

f x p px n x


Binomial Probability FunctionBinomial Probability Function

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( )!( ) (1 )

!( )!x n xn

f x p px n x

( )!( ) (1 )

!( )!x n xn

f x p px n x


!!( )!

nx n x

!!( )!

nx n x

( )(1 )x n xp p ( )(1 )x n xp p

Binomial Probability FunctionBinomial Probability Function

Probability of a particularProbability of a particular sequence of trial outcomessequence of trial outcomes with x successes in with x successes in nn trials trials

Probability of a particularProbability of a particular sequence of trial outcomessequence of trial outcomes with x successes in with x successes in nn trials trials

Number of experimentalNumber of experimental outcomes providing exactlyoutcomes providing exactly

xx successes in successes in nn trials trials

Number of experimentalNumber of experimental outcomes providing exactlyoutcomes providing exactly

xx successes in successes in nn trials trials

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Example: Evans ElectronicsExample: Evans Electronics

Evans is concerned about a low retention Evans is concerned about a low retention rate for employees. In recent years, rate 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 any the hourly employees annually. Thus, for any hourly employee chosen at random, hourly employee chosen at random, management estimates a probability of management estimates a probability of 0.1 that the person will not be with the 0.1 that the person will not be with the company next year.company next year.

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Using the Binomial Probability FunctionUsing the Binomial Probability Function

Choosing 3 hourly employees at random, Choosing 3 hourly employees at random, what is the probability that 1 of them will leave what is the probability that 1 of them will leave the company this year?the company this year?

f xn

x n xp px n x( )

!!( )!

( )( )

1f xn

x n xp px n x( )

!!( )!

( )( )

1

1 23!(1) (0.1) (0.9) 3(.1)(.81) .243

1!(3 1)!f

1 23!

(1) (0.1) (0.9) 3(.1)(.81) .2431!(3 1)!

f

LetLet: p: p = .10, = .10, nn = 3, = 3, xx = 1 = 1

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Tree DiagramTree Diagram


1st Worker 1st Worker 2nd Worker2nd Worker 3rd Worker3rd Worker xx Prob.Prob.

Leaves (.1)Leaves (.1)

Stays (.9)Stays (.9)

33

22

00

22

22



S (.9)S (.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) .0010.0010

.0090.0090

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11

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.0810.0810

.0810.0810

1111

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(1 )np p (1 )np p

EE((xx) = ) = = = npnp

Var(Var(xx) = ) = 22 = = npnp(1 (1 pp))


VarianceVariance

Standard DeviationStandard Deviation

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3(.1)(.9) .52 employees 3(.1)(.9) .52 employees

EE((xx) = ) = = 3(.1) = .3 employees out of 3 = 3(.1) = .3 employees out of 3

Var(Var(xx) = ) = 22 = 3(.1)(.9) = .27 = 3(.1)(.9) = .27


VarianceVariance

Standard DeviationStandard Deviation

Chapter 5 Discrete Probability Distributions

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