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probability_generating_functions

# probability_generating_functions - Summer 2007 STAT333...

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Summer 2007 STAT333 Summary of generating function The purpose of this part is two-folds. First, given a sequence of real numbers { a n } n =0 , how to calculate the generating function A ( s ). Secondly, given the generating function A ( s ), how to find the coeﬃcients { a n } n =0 . Generating function: definition Given a sequence of real numbers { a 0 , a 1 , . . . , a n } = { a n } n =0 , define the power series, A ( s ) = a 0 + a 1 s + a 2 s 2 + · · · = n =0 a n s n for every real number s for which the above power series converges. Obviously whether or not the power series converges at a particular s depends on the coeﬃcients { a n } n =0 . Every power series does exactly one of the following three things: (a) A ( s ) converges only for s = 0 (not our interest). (b) A ( s ) converges when | s | < R and diverges when | s | > R . In this case, R is called the convergence radius for the power series. (c) A ( s ) converges for all the real s (corresponds to the case R = ). In the last two cases, we call A ( s ) the generating function of the sequence { a n } n =0 . For the generating function, the close form for A ( s ) and the radius are both very important. When you find the close form, please do not forget the convergence radius. Examples of the commonly used generating functions (a) Geometric: A ( s ) = s k 1 s = n = k s n with radius R = 1. Here a n = 0 for n = 0 , 1 , . . . , k 1 and a n = 1 for n k. (b) Alternate Geometric: A ( s ) = ( 1) k s k 1+ s = n = k ( 1) n s n with radius R = 1. Here a n = 0 for n = 0 , 1 , . . . , k 1 and a n = ( 1) n for n k. (For the first two formulas, try to understand how you can get them by the formula for geometric series. In general, it is First term 1 common ratio .) (c) Exponential: A ( s ) = e s = n =0 s n n ! with radius R = . Here a n = 1 /n ! , for n = 0 , 1 , 2 . . . . (Try to under the above formula from n =0 s n e s n ! = 1. That is, the summation of the pmf of Poisson distribution is equal to 1.) 1

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(d) Binomial: A ( s ) = (1 + s ) a = n =0 ( a n ) s n with radius R = 1. Here a n = ( a n ) , for n = 0 , 1 , 2 . . . .
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probability_generating_functions - Summer 2007 STAT333...

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