# 02_21ans - STAT 400 Examples for Spring 2011 The k th...

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Examples for 02/21/2011 Spring 2011 The k th moment of X (the k th moment of X about the origin ), μ k , is given by μ k = E ( X k ) = ( ) x k x f x all The k th central moment of X (the k th moment of X about the mean ), μ k ' , is given by μ k ' = E ( ( X – μ ) k ) = ( ) ( ) - x k x f x all μ The moment-generating function of X, M X ( t ), is given by M X ( t ) = E ( e t X ) = ( ) x x t x f e all Theorem 1 : M X ' ( 0 ) = E ( X ) M X " ( 0 ) = E ( X 2 ) M X ( k ) ( 0 ) = E ( X k ) Theorem 2 : M X 1 ( t ) = M X 2 ( t ) for some interval containing 0 f X 1 ( x ) = f X 2 ( x ) Theorem 3 : Let Y = a X + b . Then M Y ( t ) = e b t M X ( a t ) 1. Suppose a random variable X has the following probability distribution: x f ( x ) Find the moment-generating function of X, M X ( t ). 10 0.20 11 0.40 M X ( t ) = E ( e t X ) = ( ) x x t x f e all = 0.20 e 10 t + 0.40 e 11 t + 0.30 e 12 t + 0.10 e 13 t . 12

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02_21ans - STAT 400 Examples for Spring 2011 The k th...

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