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Chapter2 - Stat1301B Probability Statistics I Chapter II...

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Stat1301B Probability& Statistics I Fall 2011-2012 P.40 Chapter II Random Variables and Probability Distributions § 2.1 Random Variables Definition A random variable Ω : X is a numerical valued function defined on a sample space. In other words, a number ( ) ω X , providing a measure of the characteristic of interest, is assigned to each outcome ω in the sample space. Remark Always keep in mind that X is a function rather than a number. The value of X depends on the outcome. We write x X = to represent the event ( ) { } | x X = Ω ω ω and x X to represent the event ( ) { } | x X Ω ω ω . Example 2.1 Let X be the number of aces in a hand of three cards drawn randomly from a deck of 52 cards. Denote A as an ace card and N as a non-ace card. Then Ω = {AAA, AAN, ANA, ANN, NAA, NAN, NNA, NNN} The space of X is {0, 1, 2, 3}. Hence X is discrete. { } 3 , 2 , 1 , 0 : Ω X such that ( ) 3 = AAA X ( ) ( ) ( ) 2 = = = NAA X ANA X AAN X ( ) ( ) ( ) 1 = = = NNA X NAN X ANN X ( ) 0 = NNN X Using simple probability calculations, we have ( ) { } ( ) 78262 . 0 3 52 3 48 0 = = = = NNN P X P ( ) { } ( ) 20416 . 0 3 52 2 48 1 4 , , 1 = × = = = NNA NAN ANN P X P ( ) { } ( ) 01303 . 0 3 52 1 48 2 4 , , 2 = × = = = NAA ANA AAN P X P ( ) { } ( ) 00018 . 0 3 52 3 4 3 = = = = AAA P X P
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