STAT 333 Handout of Probability Distributions

STAT 333 Handout of Probability Distributions - Discrete...

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Discrete Random Variables 1. Bernoulli R.V.: A Bernoulli trial is an experiment with only two possible outcomes, success ( S ) and failure ( F ). Bernoulli trials are defined as being independent trials, each with a constant probability of success P ( S ) = p . The Bernoulli random variable is given by X := I j = ( 1 , if outcome is S 0 , if outcome is F . range { 0 , 1 } p.m.f f (0) = P ( X = 0) = 1 - p , f (1) = P ( X = 1) = p mean E ( X ) = p variance V ar ( X ) = p (1 - p ) p.g.f. G X ( s ) = 1 - p + ps for s R 2. Binomial R.V.: X : total number of successes in n Bernoulli trials. X := I 1 + I 2 + ··· + I n . X BIN ( n,p ) with parameters n ( n = 1 , 2 , 3 ,... ) and p (0 p 1) range { 0 , 1 ,...,n } p.m.f f ( k ) = P ( X = k ) = ± n k ) p k (1 - p ) n - k mean E ( X ) = np variance V ar ( X ) = np (1 - p ) p.g.f. G X ( s ) = (1 - p + ps ) n for s R 3. Geometric R.V.: Y : number of trials up to and including the first success in repeated Bernoulli trials. Y
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This note was uploaded on 01/21/2012 for the course STAT 333 taught by Professor Chisholm during the Fall '08 term at Waterloo.

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STAT 333 Handout of Probability Distributions - Discrete...

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