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# lect10 - THE CONDITIONAL RULE OF EXPECTATION SOME MORE...

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THE CONDITIONAL RULE OF EXPECTATION SOME MORE COMMON DISCRETE DISTRIBUTIONS Outline THE CONDITIONAL RULE OF EXPECTATION SOME MORE COMMON DISCRETE DISTRIBUTIONS Geometric Distributions Uniform Distributions Hypergeometric Disributions Negative Binomial Distributions An application: How long does a game of craps last? 1 / 14 Xinghua Zheng Lect 10: Expectations and variances: examples and methods

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THE CONDITIONAL RULE OF EXPECTATION SOME MORE COMMON DISCRETE DISTRIBUTIONS THE CONDITIONAL RULE OF EXPECTATION The Conditioning Rule. Let C 1 , C 2 , . . . be a collection of mutually exclusive and exhaustive cases. For each i , let E ( X | C i ) = X x x · P [ X = x | C i ] . be the conditional expectation of X given case C i . Then E ( X ) = X i P [ C i ] · E ( X | C i ) . 2 / 14 Xinghua Zheng Lect 10: Expectations and variances: examples and methods
THE CONDITIONAL RULE OF EXPECTATION SOME MORE COMMON DISCRETE DISTRIBUTIONS Geometric Distributions Uniform Distributions Hypergeometric THE GEOMETRIC DISTRIBUTIONS Let N be the number of tosses required to produce the first head, when tossing a coin that has probability p to land heads. N has a geometric distribution with success probability p What is the expected value of N ? Method 1: By definition, E ( N ) = n = 1 n · q n - 1 p = . . . . Method 2: Let H be the event that the first toss is a head, and T = H c the event that first toss is a tail. Then P [ H ] = , P [ T ] = , E ( N | H ) = , E ( N | T ) = , E ( N ) = = , ( 1 - q ) E ( N ) = , E ( N ) = . 3 / 14 Xinghua Zheng Lect 10: Expectations and variances: examples and methods

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THE CONDITIONAL RULE OF EXPECTATION SOME MORE COMMON DISCRETE DISTRIBUTIONS Geometric Distributions Uniform Distributions Hypergeometric THE GEOMETRIC DISTRIBUTIONS, ctd Similarly we can find its variance: Put X = N ( N - 1 ) = N 2 - N . Then E ( X | H ) = , E ( X | T ) = , E ( N 2 ) - E ( N ) = E ( X ) = = q ( E ( N 2 ) + E ( N )) , E ( N 2 ) = ( 1 + q ) E ( N ) / p = ( 1 + q ) / p 2 , Var ( N ) = q p 2 , SD ( N ) = q p .
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• Spring '11
• Zheu
• Probability theory, Binomial distribution, Geometric distribution, Common Discrete Distributions, Xinghua Zheng

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lect10 - THE CONDITIONAL RULE OF EXPECTATION SOME MORE...

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