STAT 400 Lecture AL1 Examples for 2 3 Spring 2015 Dalpiaz The k th moment of X the k th moment of X about the origin k is given by k E X k x k f x all x The k th central moment of X the k th moment of X about the mean k is given by k E X k x k f x all x The moment generating function of X M X t is given by MX t E et X e t x f x all x Theorem 1 M X 0 E X M X 0 E X 2 k MX 0 E X k Theorem 2 M X t M X t for some interval containing 0 1 2 Theorem 3 1 fX1 x fX2 x Let Y a X b Then M Y t e b t M X a t Suppose a random variable X has the following probability distribution x f x 10 0 20 11 0 40 12 0 30 13 0 10 Find the moment generating function of X M X t 2 Suppose the moment generating function of a random variable X is t 2t 0 25 e 3 t 0 30 e 5 t M X t 0 10 0 15 e 0 20 e Find the expected value of X E X 3 Suppose a discrete random variable X has the following probability distribution f 0 P X 0 2 e 1 2 f k P X k 1 k 1 2 3 2 k k a Find the moment generating function of X M X t b Find the expected value of X E X and the variance of X Var X 4 Let X be a Binomial n p random variable Find the moment generating function of X 5 Let X be a geometric random variable with probability of success p a Find the moment generating function of X b Use the moment generating function of X to find E X 6 a Find the moment generating function of a Poisson random variable Consider ln M X t ln M X t Since M X t MX t cumulant generating function ln M X t M X t M X t M X t 2 MX t M X 0 1 M X 0 E X M X 0 E X 2 ln M X t t 0 E X X ln M X t t 0 E X 2 E X 2 X2 b Find E X and Var X where X is a Poisson random variable 2
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