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,Click to Edit Master Title Style,Click to edit Master text styles,Second Level,Third Level,1,Slide,2011 Cengage Learning.All Rights Reserved.May not be scanned,copied,or duplicated,or posted to a publicly accessible website,in whole or in part.,Statistics for Businessand Economics,Anderson Sweeney Williams,Slides by,John Loucks,St.Edwards University,Statistics for Businessand Ec,Chapter 5 Discrete Probability Distributions,.10,.20,.30,.40,0,1,2 3 4,Random Variables,Discrete Probability Distributions,Expected Value and Variance,Binomial Probability Distribution,Poisson Probability Distribution,Hypergeometric Probability Distribution,Chapter 5 Discrete Probabilit,A,random variable,is a numerical description of the,outcome of an experiment.,Random Variables,A,discrete random variable,may assume either a,finite number of values or an infinite sequence of,values.,A,continuous random variable,may assume any,numerical value in an interval or collection of,intervals.,A random variable is a numeri,Let,x,=number of TVs sold at the store in one day,where,x,can take on 5 values(0,1,2,3,4),Example:JSL Appliances,Discrete Random Variable,with a Finite Number of Values,We can count the TVs sold,and there is a finite,upper limit on the number that might be sold(which,is the number of TVs in stock).,Let x=number of TVs sold at,Let,x,=number of customers arriving in one day,where,x,can take on the values 0,1,2,.,Discrete Random Variable,with an Infinite Sequence of Values,We can count the customers arriving,but there is,no finite upper limit on the number that might arrive.,Example:JSL Appliances,Let x=number of customers a,Random Variables,Question,Random Variable,x,Type,Family,size,x,=Number of dependents,reported on tax return,Discrete,Distance from,home to store,x,=Distance in miles from,home to the store site,Continuous,Own dog,or cat,x,=1 if own no pet;,=2 if own dog(s)only;,=3 if own cat(s)only;,=4 if own dog(s)and cat(s),Discrete,Random VariablesQuestionRandom,The,probability distribution,for a random variable,describes how probabilities are distributed over,the values of the random variable.,We can describe a discrete probability distribution,with a table,graph,or formula.,Discrete Probability Distributions,The probability distribution,The probability distribution is defined by a,probability function,denoted by,f,(,x,),which provides,the probability for each value of the random variable.,The required conditions for a discrete probability,function are:,Discrete Probability Distributions,f,(,x,),0,f,(,x,)=1,The probability distribution,a,tabular representation,of the probability,distribution for TV sales was developed.,Using past data on TV sales,Number,Units Sold,of Days,0 80,1 50,2 40,3 10,4,20,200,x,f,(,x,),0 .40,1 .25,2 .20,3 .05,4,.10,1.00,80/200,Discrete Probability Distributions,Example:JSL Appliances,a tabular representation of,.10,.20,.30,.,40,.50,0,1 2,3,4,Values of Random Variable,x,(TV sales),Probability,Discrete Probability Distributions,Example:JSL Appliances,Graphical,representation,of probability,distribution,.10.20.30.40.500 1 2,Discrete Uniform Probability Distribution,The,discrete uniform probability distribution,is the,simplest example of a discrete probability,distribution given by a formula.,The,discrete uniform probability function,is,f,(,x,)=1/,n,where:,n,=the number of values the random,variable may assume,the values of the,random variable,are equally likely,Discrete Uniform Probability D,Expected Value,The,expected value,or mean,of a random variable,is a measure of its central location.,The expected value is a weighted average of the,values the random variable may assume.The,weights are the probabilities.,The expected value does not have to be a value the,random variable can assume.,E,(,x,)=,=,xf,(,x,),Expected Value The expected va,Variance and Standard Deviation,The,variance,summarizes the variability in the,values of a random variable.,The variance is a weighted average of the squared,deviations of a random variable from its mean.The,weights are the probabilities.,Var(,x,)=,2,=,(,x,-,),2,f,(,x,),The,standard deviation,is defined as the positive,square root of the variance.,Variance and Standard Deviatio,expected number of TVs sold in a day,x,f,(,x,),xf,(,x,),0 .40 .00,1 .25 .25,2 .20 .40,3 .05 .15,4 .10,.40,E,(,x,)=1.20,Expected Value,Example:JSL Appliances,expected number of TVs sold in,0,1,2,3,4,-1.2,-0.2,0.8,1.8,2.8,1.44,0.04,0.64,3.24,7.84,.40,.25,.20,.05,.10,.576,.010,.128,.162,.784,x-,(,x-,),2,f,(,x,),(,x,-,),2,f,(,x,),Variance of daily sales=,s,2,=1.660,x,TVs,squared,Standard deviation of daily sales=1.2884 TVs,Variance,Example:JSL Appliances,0-1.21.44.40.576x-(x-)2f,Binomial Probability Distribution,Four Properties of a Binomial Experiment,3.The probability of a success,denoted by,p,does,not change from trial to trial.,4.The trials are independent.,2.Two outcomes,success,and,failure,are possible,on each trial.,1.The experiment consists of a sequence of,n,identical trials.,stationa
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