l12_rvs - Random Variables Reading Ross Ch 4 Sec 1-2 Monday...

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03/22/2009 CS206 - Intro. to Discrete Structures II 1 Random Variables Reading: Ross, Ch 4., Sec. 1-2. Monday, March 23, 2009
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03/22/2009 CS206 - Intro. to Discrete Structures II 2 Gambler’s ruin Recall gambler’s ruin problem. Consider player A’s profit after k=3 coin tosses: Outcome P(O) Profit HHH 1/8 3 HHT 1/8 1 HTH 1/8 1 HTT 1/8 -1 THH 1/8 1 THT 1/8 -1 TTH 1/8 -1 TTT 1/8 -3 Monday, March 23, 2009
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03/22/2009 CS206 - Intro. to Discrete Structures II 3 Function of outcomes The players are not necessarily interested in outcomes, they are interested in profit. Outcomes associated with profit: A 1 = “profit 3” = { HHH } P(A 1 )=1/8 A 2 = “profit 1” = { HHT, HTH, THH } P(A 2 )=3/8 A 3 = “profit -1” = { TTH, THT, HTT } P(A 3 )=3/8 A 4 = “profit -3” = { TTT } P(A 4 )=1/8 Monday, March 23, 2009
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03/22/2009 CS206 - Intro. to Discrete Structures II 4 Random variable Random variable is a FUNCTION that maps events to the set of (in general) real numbers. X: Ω E.g., X = Profit is a random variable: Domain = Ω Range = {-3,-1,1,3} Profit( {HHT} ) = 1 Profit P(Profit=x) 3 1/8 1 3/8 -1 3/8 -3 1/8 Monday, March 23, 2009
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03/22/2009 CS206 - Intro. to Discrete Structures II 5 Examples of RV’s Experiment Outcomes (sample space Ω ) | Ω | Random variable X
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