# 09_13_2 - e 2 t + 0.25 e 3 t + 0.30 e 5 t . Find the...

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STAT 400 Examples for 09/13/2011 (2) Fall 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
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Unformatted text preview: 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 ( ) = 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 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 )....
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## 09_13_2 - e 2 t + 0.25 e 3 t + 0.30 e 5 t . Find the...

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