STAT 400 Lecture AL1 Spring 2015 Dalpiaz Examples for 2 3 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 MX t E et X e t x f x all x 0 20 e 10 t 0 40 e 11 t 0 30 e 12 t 0 10 e 13 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 M X t 0 15 e t 0 40 e 2 t 0 75 e 3 t 1 50 e 5 t E X M X 0 0 15 0 40 0 75 1 50 1 30 OR 3 x f x x f x 0 0 10 0 1 0 15 0 15 2 0 20 0 4 3 0 25 0 75 5 0 30 1 5 E X 1 30 all x Suppose a discrete random variable X has the following probability distribution f 0 P X 0 2 e 1 2 a x f x f k P X k 1 k 1 2 3 2 k k Find the moment generating function of X M X t MX t e t x f x 1 2 e 1 2 k 1 all x e t k k1 2 k 2 e1 2 k 1 t 1 e1 2 e e 2 e t 2 k k 2 e1 2 et 2 e 1 b Find the expected value of X E X and the variance of X Var X t M X t e e 2 t e e t MX 2 et 2 t e 2 E X 2 M X 0 Var X E X 2 E X M X 0 e 2 3 ee t 2 2 t e 2 e1 2 4 3 E X 2 e1 2 4 4 12 1 e 4 Let X be a Binomial n p random variable Find the moment generating function of X n MX t k 0 n p k 1 p n k k e t k n k n p e t 1 p n k k k 0 1 p p e t n 5 Let X be a geometric random variable with probability of success p a Find the moment generating function of X MX t e t k 1 p k 1 p p e t e t k 1 1 p k 1 k 1 k 1 p e t n 0 1 p e t n p e t 1 1 p e t t ln 1 p b Use the moment generating function of X to find E X MX t p e t 1 1 p e t 6 a 2 1 1 p e t t t p e 1 p e 1 1 p e t 2 p e t E X MX 0 t ln 1 p p 1 p 2 p Find the moment generating function of a Poisson random variable k e MX t etk k k 0 e e e t e e t e e t k 0 1 k k 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 ln M X t e t 1 ln M X t e t ln M X t t 0 E X ln M X t e t ln M X t t 0 Var X
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