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Unformatted text preview: 2. Random Variables Dave Goldsman Georgia Institute of Technology, Atlanta, GA, USA 7/7/11 Goldsman 7/7/11 1 / 144 Outline 1 Intro / Definitions 2 Discrete Random Variables 3 Continuous Random Variables 4 Cumulative Distribution Functions 5 Great Expectations 6 Functions of a Random Variable 7 Bivariate Random Variables 8 Conditional Distributions 9 Independent Random Variables 10 Conditional Expectation 11 Covariance and Correlation 12 Moment Generating Functions Goldsman 7/7/11 2 / 144 Intro / Definitions Definition: A random variable (RV) is a function from the sample space to the real line. X : S → < . Example: Flip 2 coins. S = { HH,HT,TH,TT } . Suppose X is the RV corresponding to the # of H ’s. X ( TT ) = 0 , X ( HT ) = X ( TH ) = 1 , X ( HH ) = 2 . P ( X = 0) = 1 4 , P ( X = 1) = 1 2 , P ( X = 2) = 1 4 . Notation: Capital letters like X,Y,Z,U,V,W usually represent RV’s. Small letters like x,y,z,u,v,w usually represent particular values of the RV’s. Thus, you can speak of P ( X = x ) . Goldsman 7/7/11 3 / 144 Intro / Definitions Example: Let X be the sum of two dice rolls. Then X ((4 , 6)) = 10 . In addition, P ( X = x ) = 1 / 36 if x = 2 2 / 36 if x = 3 . . . 6 / 36 if x = 7 . . . 1 / 36 if x = 12 otherwise Goldsman 7/7/11 4 / 144 Intro / Definitions Example: Flip a coin. X ≡ ( if T 1 if H Example: Roll a die. Y ≡ ( if { 1 , 2 , 3 } 1 if { 4 , 5 , 6 } For our purposes, X and Y are the same, since P ( X = 0) = P ( Y = 0) = 1 2 and P ( X = 1) = P ( Y = 1) = 1 2 . Goldsman 7/7/11 5 / 144 Intro / Definitions Example: Select a real # at random betw 0 and 1. Infinite number of “equally likely” outcomes. Conclusion: P ( each individual point ) = P ( X = x ) = 0 , believe it or not! But P ( X ≤ . 5) = 0 . 5 and P ( X ∈ [0 . 3 , . 7]) = 0 . 4 . If A is any interval in [0,1], then P ( A ) is the length of A . Goldsman 7/7/11 6 / 144 Intro / Definitions Definition: If the number of possible values of a RV X is finite or countably infinite, then X is a discrete RV. Otherwise,. . . A continuous RV is one with prob 0 at every point. Example: Flip a coin — get H or T . Discrete. Example: Pick a point at random in [0 , 1] . Continuous. Goldsman 7/7/11 7 / 144 Discrete Random Variables Outline 1 Intro / Definitions 2 Discrete Random Variables 3 Continuous Random Variables 4 Cumulative Distribution Functions 5 Great Expectations 6 Functions of a Random Variable 7 Bivariate Random Variables 8 Conditional Distributions 9 Independent Random Variables 10 Conditional Expectation 11 Covariance and Correlation 12 Moment Generating Functions Goldsman 7/7/11 8 / 144 Discrete Random Variables Definition: If X is a discrete RV, its probability mass function (pmf) is f ( x ) ≡ P ( X = x ) ....
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This note was uploaded on 08/27/2011 for the course ISYE 3770 taught by Professor Goldsman during the Spring '07 term at Georgia Tech.
 Spring '07
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