chapter 5 discrete probability distributions

Chapter 5Chapter 5 Discrete Probability Distributions Discrete Probability Distributions

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

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.20.20

.30.30

.40.40

0 1 2 3 4 0 1 2 3 4

Random VariablesRandom Variables

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

A random variable can be classified as being A random variable can be classified as being either discrete or continuous depending on the either discrete or continuous depending on the numerical values it assumes.numerical values it assumes.

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

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

Example: JSL AppliancesExample: JSL Appliances

Discrete random variable with a finite number of Discrete random variable with a finite number of valuesvalues

Let Let xx = number of TV sets sold at the store in one = number of TV sets sold at the store in one dayday

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

Discrete random variable with an infinite sequence Discrete random variable with an infinite sequence of valuesof values

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, . . .

We can count the customers arriving, but there is no We can count the customers arriving, but there is no finite upper limit on the number that might arrive.finite upper limit on the number that might arrive.

Random VariablesRandom Variables

QuestionQuestion Random Variable Random Variable xx Type Type

Family Family xx = Number of dependents in Discrete = Number of dependents in Discrete

size family reported on tax return size family reported on tax return

Distance from Distance from xx = Distance in miles from = Distance in miles from Continuous Continuous

home to store home to the store site home to store home to the store site

Own dog Own dog xx = 1 if own no pet; = 1 if own no pet; Discrete Discrete

or cat or cat = 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)

Discrete Probability DistributionsDiscrete Probability Distributions

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

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

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

ff((xx) ) >> 0 0

ff((xx) = 1) = 1 We can describe a discrete probability We can describe a discrete probability

distribution with a table, graph, or equation.distribution with a table, graph, or equation.

Using past data on TV sales (below left), a Using past data on TV sales (below left), a tabular representation of the probability tabular representation of the probability distribution for TV sales (below right) was distribution for TV sales (below right) was developed.developed.

NumberNumber

Units SoldUnits Sold of Daysof Days xx ff((xx))

00 80 80 0 0 .40 .40

11 50 50 1 1 .25 .25

22 40 40 2 2 .20 .20

33 10 10 3 3 .05 .05

44 2020 4 4 .10 .10

200200 1.00 1.00



Graphical Representation of the Probability Graphical Representation of the Probability DistributionDistribution

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.20.20

.30.30

.40.40

.50.50

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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Discrete Uniform Probability DistributionDiscrete Uniform Probability Distribution

The The discrete uniform probability distributiondiscrete uniform probability distribution is is the simplest example of a discrete probability the simplest example of a discrete probability distribution given by a formula.distribution given by a formula.

The The discrete uniform probability functiondiscrete uniform probability function is is

ff((xx) = 1/) = 1/NN

where:where:

NN = the number of values the = the number of values the randomrandom

variable may assumevariable may assume Note that the values of the random variable Note that the values of the random variable

are equally likely.are equally likely.

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

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

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

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

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

Expected Value and VarianceExpected Value and Variance


Expected Value of a Discrete Random VariableExpected Value of a Discrete Random Variable

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

The expected number of TV sets sold in a day The expected number of TV sets sold in a day is 1.2is 1.2

Variance and Standard DeviationVariance and Standard Deviation

of a Discrete Random Variableof a Discrete Random Variable

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

00 -1.2-1.2 1.44 1.44 .40.40 .576 .57611 -0.2-0.2 0.04 0.04 .25.25 .010 .01022 0.8 0.8 0.64 0.64 .20.20 .128 .12833 1.8 1.8 3.24 3.24 .05.05 .162 .16244 2.8 2.8 7.84 7.84 .10.10 .784 .784

1.660 = 1.660 =

The variance of daily sales is 1.66 TV sets The variance of daily sales is 1.66 TV sets squaredsquared.. The standard deviation of sales is 1.2884 TV The standard deviation of sales is 1.2884 TV sets.sets.


The supervisor of XYZ would like to know what the The supervisor of XYZ would like to know what the average number of absentees is daily and also what the average number of absentees is daily and also what the standard deviation is of the daily employee absentee standard deviation is of the daily employee absentee rate. Historical data provides the follow probability rate. Historical data provides the follow probability distribution:distribution:

Daily Daily

AbsenteesAbsentees ff((xx))

00 .50 .50

11 .23 .23

22 .12 .12

33 .10 .10

44 .02 .02

55 .02 .02

66 .01.01

1.001.00

Example: XYZ ElectronicsExample: XYZ Electronics


Expected Value of a Discrete Random VariableExpected Value of a Discrete Random Variable

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

00 .50 .50 .00 .00

11 .23 .23 .23 .23

22 .12 .12 .24 .24

33 .10 .10 .30 .30

44 .02 .02 .08 .08 55 .02 .02 .10 .10 66 .01 .01 .06.06

EE((xx) = 1.01) = 1.01

The expected number of absentees is 1.01The expected number of absentees is 1.01

Variance and Standard DeviationVariance and Standard Deviation

of a Discrete Random Variableof a Discrete Random Variable

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

00 -1.01-1.01 1.0201 1.0201 .50.50 .5101 .510111 -0.01-0.01 0.0001 0.0001 .23.23 .0000 .000022 0.99 0.99 0.9801 0.9801 .12.12 .1176 .117633 1.99 1.99 3.9601 3.9601 .10.10 .3960 .396044 2.99 2.99 8.9401 8.9401 .02.02 .1788 .178855 3.99 3.99 15.9201 15.9201 .02.02 .3184 .318466 4.99 4.99 24.9001 24.9001 .01.01 .2490.2490

1.7699 = 1.7699 =

The variance of absentees is 1.7699 The variance of absentees is 1.7699 squaredsquared..

The standard deviation of absentees is 1.3304.The standard deviation of absentees is 1.3304.


chapter 5 discrete probability distributions

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