MATH 464 HOMEWORK 8 SPRING 2013 The following assignment is to be turned in on Thursday, April 4, 2013. 1. Let X be a Poisson random variable with parameter > 0. a) Find the moment generating function for X. b) Use your result above to nd the mean of the random variable Z = 2X3 ?? 3X2 + X. c) Consider n 1, independent, discrete random variables X1, X2, , Xn, and suppose that each are Poisson with parameter > 0. Let Z = X1 + X2 + + Xn. Find the pmf of Z. 2. Let X be a negative binomial random variable with parameters n and p. Calculate the variance of X. 3. Let X be an exponential random variable with parameter > 0. a) Let t 0 and calculate P(X t). b) Let s; t 0 and calculate P(X s + tjX s). (You can compare your answer to this question with your answer to problem #5 on homework #5.) 4. The gamma function is dened by ??(w) = Z 1 0 xw??1e??x dx for all w > 0. In terms of this function, a continuous random variable X (with parameters w > 0 and > 0) is dened by setting fX(x) = w ??(w)xw??1e??x if x > 0; 0 otherwise: and declaring that X has probability density function fX(x). (fX is called the gamma distribution with parameters w > 0 and > 0.) a) Show that X is a continuous random variable by showing that ZR fX(t) dt = 1 for all values of w > 0 and > 0. 1 2 SPRING 2013 b) Show that for any w > 1, ??(w) = (w ?? 1)??(w ?? 1) Use your result to calculate ??(n) for any integer n 2. c) Compute the mean and variance of this random variable X.

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