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 GEO ( p ) with parameter p (0 p 1) range { 1 , 2 , . . . } p.m.f f ( k ) = P ( Y = k ) = (1 - p ) k - 1 p mean E ( Y ) = 1 p variance V ar ( Y ) = 1 - p p 2 p.g.f. G Y ( s ) = ps 1 - s (1 - p ) for | s | < 1 1 - p 4. Negative Binomial R.V.: Y
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