elemprob-fall2010-page8

elemprob-fall2010-page8 - To do that, start with ∞ xn =...

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Unformatted text preview: To do that, start with ∞ xn = n=0 Differentiating, ∞ nxn−1 = n=1 1 . 1−x 1 . (1 − x)2 Letting x = 1 − p, we obtain EX = 1 1 p= . (1 − (1 − p))2 p This is reasonable: the number of tosses to get a head should be about 1/p. For a “real-life” example, in China, parents are allowed to have only one child. However, in rural areas, if the child is a girl, they are allowed to have a second child. Suppose the rule were that parents could have children until they had a boy. How many girls would a family have on average? The number of children per family has a geometric distribution with p = 1/2, so the expected number of children is 2; one of them is a boy, so the expected number of girls is 1. The quantity E X k is called the k th moment of X , and the first moment is also called the mean. If we roll a die, E X 2 = 1 · 1 + 4 · 1 + · · · + 36 · 6 6 1 6 = 91 . 6 Make the observation that if X is equal to a with probability one, then E X = a. Proposition 3.4 E (X + a) = E X + a and E (aX ) = aE X. 8 ...
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This note was uploaded on 12/29/2011 for the course MATH 316 taught by Professor Ansan during the Spring '10 term at SUNY Stony Brook.

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