# week11 - week 11 1 Some facts about Power Series ∞ •...

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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, π X (1) = p + p 1 + p 2 +… = 1....
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week11 - week 11 1 Some facts about Power Series ∞ •...

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