1 so the median is always close to the mean np 1 r

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Unformatted text preview: ll that the Taylor series expansion of a function f about a point xo is given by f (x) = f (xo ) + f (xo )(x − xo ) + f (xo ) (x − xo )2 (x − xo )3 + f (xo ) + ··· . 2 3! The MacLaurin series expansion of a function f is the Taylor series expansion about xo = 0 : f (x) = f (0) + f (0)x + f (0) x3 x2 + f (0) + · · · . 2 3! The MacLaurin series expansion of f (x) = (1 + x)n is given by n (1 + x)n = k=0 nk x. k Substituting x = p/(1 − p) into (2.3) and multiplying through by (1 − p)n yields that n k=0 nk p (1 − p)n−k = 1, k (2.3) 2.4. INDEPENDENCE AND THE BINOMIAL DISTRIBUTION 35 so the binomial pmf sums to one, as expected. Since each trial results in a one with probability p, and there are n trials, the mean number of ones is given by E [X ] = np. This same result can be derived with more work from the pmf: n k nk p (1 − p)n−k k k E [X ] = nk p (1 − p)n−k k k=0 n = k=1 n = np k=1 n−1 (n − 1)! pk−1 (1 − p)n−k (n − k )!(k − 1)! = np l=0 n−1 l p (1 − p)n−1−l l (here l = k − 1) = np. (2.4) The variance of the...
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This note was uploaded on 02/09/2014 for the course ISYE 2027 taught by Professor Zahrn during the Spring '08 term at Georgia Institute of Technology.

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