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

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THE CONDITIONAL RULE OF EXPECTATION 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 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
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THE CONDITIONAL RULE OF EXPECTATION Geometric Distributions Uniform Distributions Hypergeometric D 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 Geometric Distributions Uniform Distributions Hypergeometric D 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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This note was uploaded on 11/27/2011 for the course ISOM 3540 taught by Professor Zheu during the Spring '11 term at HKUST.

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

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