Chapter 5 Discrete Probability DistributionsRandom VariablesExample: JSL AppliancesSlide 4Discrete Probability DistributionsSlide 6Slide 7Discrete Uniform Probability DistributionExpected Value and VarianceSlide 10Slide 11Example: XYZ ElectronicsExample: XYZ ElectronicsSlide 14Chapter 5Chapter 5 Discrete Probability Distributions Discrete Probability DistributionsRandom VariablesRandom VariablesDiscrete Probability DistributionsDiscrete Probability DistributionsExpected Value and VarianceExpected Value and Variance.10.10.20.20.30.30.40.40 0 1 2 3 4 0 1 2 3 4Random 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 valuesvaluesLet Let xx = number of TV sets sold at the store in = number of TV sets sold at the store in one dayone day 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 Discrete random variable with an infinite sequence of valuessequence of valuesLet 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 We can count the customers arriving, but there is no finite upper limit on the number that might is no finite upper limit on the number that might arrive.arrive.Random VariablesRandom VariablesQuestionQuestion Random Variable Random Variable xx Type Type Family Family xx = Number of dependents in Discrete = Number of dependents in Discretesize family reported on tax return size family reported on tax return 77Distance from Distance from xx = Distance in miles from = Distance in miles from Continuous Continuoushome 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 Discreteor 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 variable for a random variable describes how probabilities are distributed over describes how probabilities are distributed over the values of the random variable.the values of the random 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 probability The required conditions for a discrete probability function are: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 1.00Example: JSL AppliancesExample: JSL AppliancesExample: JSL AppliancesExample: JSL AppliancesGraphical Representation of the Probability Graphical Representation of the Probability DistributionDistribution.10.10.20.20.30.30.40.40.50.500 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)ProbabilityProbabilityProbabilityProbabilityDiscrete 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/NNwhere:where: NN = the number of values the random = the number of values the random 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