week11 - week 11 1 Some facts about Power Series Consider...

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Unformatted text preview: week 11 1 Some facts about Power Series Consider the power series with non-negative coefficients a k . If converges for any positive value of t , say for t = r , then it converges for all t in the interval [-r, r ] and thus defines a function of t on that interval. For any t in (-r, r ), this function is differentiable at t and the series converges to the derivatives. Example: For k = 0, 1, 2, and -1< x < 1 we have that (differentiating geometric series). = k k k t a = k k k t a = 1 k k k t ka ( ) = + = 1 1 m m k x k m k x week 11 2 Generating Functions For a sequence of real numbers { a j } = a , a 1 , a 2 ,, the generating function of { a j } is if this converges for | t | < t for some t > 0. ( ) = = j j j t a t A week 11 3 Probability Generating Functions Suppose X is a random variable taking the values 0, 1, 2, (or a subset of the non-negative integers). L e t p j = P ( X = j ) , j = 0, 1, 2, . This is in fact a sequence p , p 1 , p 2 , Definition: The probability generating function of X is Since if | t | < 1 and the pgf converges absolutely at least for | t | < 1. In general, X (1) = p + p 1 + p 2 + = 1....
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week11 - week 11 1 Some facts about Power Series Consider...

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