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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 momentgenerating function of X. 5. Let X be a geometric random variable with probability of “success” p . a) Find the momentgenerating function of X. b) Use the momentgenerating function of X to find E ( X ). 6. a) Find the momentgenerating 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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 Spring '08
 Kim
 Variance, Probability theory, 2 K, µk, momentgenerating function

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