ProbabilityGeneratingFunctions

ProbabilityGeneratingFunctions - STAT 333 Probability...

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Unformatted text preview: STAT 333 Probability Generating Functions Definition: Let X have range { , 1 , 2 ,... }{} and let p n = P ( X = n ) for n = 0 , 1 , 2 ,... . We define the probability generating function (pgf) of X to be G X ( s ) = n =0 p n s n . Note that the pgf of X is just the generating function of the sequence of probabilities p ,p 1 ,p 2 ,... . G X (1) = n =0 p n 1. Thus the radius R of convergence of G X ( s ) is at least 1. X is proper if G X (1) = 1, and X is improper if G X (1) < 1. if X is proper then G X ( s ) = E ( s X ) = n =0 s n P ( X = n ) for all- R < s < R . Examples: 1. Let X binom( n,p ). Then G X ( s ) = E ( s X ) = n k =0 s k P ( X = k ) = n k =0 s k n k p k (1- p ) n- k = n k =0 n k ( ps ) k (1- p ) n- k = (1- p + ps ) n applying the binomial theorem Here the radius of convergence is R = . 2. Let X geometric( p ). Then G X ( s ) = E ( s X ) = k =1 s k (1- p ) k- 1 p = p 1- p k =1 [ s (1- p )] k = ps 1- s (1- p ) for | s (1- p ) | <...
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ProbabilityGeneratingFunctions - STAT 333 Probability...

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