4.DiscreteRandomVariablesandProbablityDistribution

# 4.DiscreteRandomVariablesandProbablityDistribution -...

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Discrete Random Variables and Probability Distributions GE331/IE300 Gülser Köksal UIUC, IESE, 2011

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G.Köksal, UIUC, 2011 2 Outline • Random variables • Probability distribution • Probability mass function • Discrete probability distributions
G.Köksal, UIUC, 2011 3 Random Variables • A random variable is a numerical description of the outcome of a random experiment. • A discrete random variable may assume either a finite number of values or an infinite sequence of values. – E.g. The number of cars arriving at a tollbooth during a one- day period • A continuous random variable may assume any numerical value in an interval or collection of intervals. – E.g.Moisture content of a package measured in percentages

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G.Köksal, UIUC, 2011 4 Examples of Discrete Random Variables Experiment Random Variable, X Possible values, x Contact five customers Number of customers who place an order 0,1,2,3,4,5 Inspect a shipment of 50 radios Number of defective radios 0,1,2,. ..,49,50 Operate a tollbooth for one day Number of cars arriving at the tollbooth 0,1,2,3,. .. Inspect a textiles product Test result 0 if failed; 1 if passed
G.Köksal, UIUC, 2011 5 Probability Distribution • A probability distribution is a description of the probabilities associated with possible values of a random variable X – Discrete r.v.: probability mass function ( pmf ) – Continuous r.v.: probability density function ( pdf ) – Cumulative distribution function ( cdf ) – Other ways to describe probability distribution

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G.Köksal, UIUC, 2011 6 A Discrete Probability Distribution Example X = sum of numbers on two dice P(X=2) =1/36 P(X=3) =2/36 P(X=7)=6/36 P(X=11)=2/36 P(X=12)=1/36
G.Köksal, UIUC, 2011 7 The Probability Mass Function of the Example x

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G.Köksal, UIUC, 2011 8 Probability Mass Function • For discrete r.v. X with possible values x 1 ,…,x n , the pmf is defined by n i i i i i x f x f x X P x f 1 1 ) ( 0 ) ( ) ( ) (
G.Köksal, UIUC, 2011 9 pmf Example Example: In a batch of 100 bottles, 5 are defective. Two bottles are randomly selected. X is the number of defective bottles. What is the probability distribution of X ? P(X=

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