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# 02_21 - X M X t b Find the expected value of X E X and the...

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STAT 400 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 12 0.30 13 0.10

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2. Suppose the moment-generating function of a random variable X is M X ( t ) = 0.10 + 0.15 e t + 0.20 e 2 t + 0.25 e 3 t + 0.30 e 5 t . 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 1 2 e - , f ( k ) = P( X = k ) = ! 2 1 k k , k = 1, 2, 3, … a) Find the moment-generating function of X , M X ( t ) . b) Find the expected value of

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Unformatted text preview: 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 ). ( cumulant generating function ) ( ln M X ( t ) ) ' = ( ) ( ) X X M M ' t t ( ln M X ( t ) ) " = ( ) ( ) ( ) ( ) [ ] 2 X 2 X X X M M M M ' " t t t t -⋅ Since M X ( ) = 1, M X ' ( ) = E ( X ), M X " ( ) = E ( X 2 ), ( ln M X ( t ) ) ' | t = 0 = E ( X ) = μ X ( ln M X ( t ) ) " | t = 0 = E ( X 2 ) – [ E ( X ) ] 2 = σ X 2 b) Find E ( X ) and Var ( X ), where X is a Poisson random variable....
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02_21 - X M X t b Find the expected value of X E X and the...

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