# Hw02ans - STAT 410 Spring 2008 Homework#2(due Friday...

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Unformatted text preview: STAT 410 Spring 2008 Homework #2 (due Friday, February 1, by 3:00 p.m.) 1. Below is a list of moment-generating functions. Provide the mean and variance of the random variable associated with each. ( Justify your answers. ) a) M X ( t ) = 10 5 3 5 2 & & ¡ ¢ £ £ ¤ ¥ + t e . Binomial, n = 10, p = 5 2 = 0.40. E ( X ) = n p = 4, Var ( X ) = n p ( 1 – p ) = 2.4. b) M X ( t ) = 6 2 . 1 1 & ¡ ¢ £ ¤ ¥- t , t < 5. Gamma, α = 6, θ = 0.20. E ( X ) = α θ = 1.2, Var ( X ) = α θ 2 = 0.24. c) M X ( t ) = 6 2 . 1 8 . & & & & & ¡ ¢ £ £ £ £ £ ¤ ¥- t t e e , t < ln 5. Negative Binomial, r = 6, p = 0.80. E ( X ) = p r = 7.5, Var ( X ) = ( ) 2 1 p p r- = 1.875. d) M X ( t ) = e 5 t . P ( X = 5 ) = 1. E ( X ) = 5, Var ( X ) = 0. 2. Let Y denote a random variable with probability density function given by f ( y ) = 2 1 y e- , – ∞ < y < ∞ . a) Find the moment-generating function of Y. ( Be sure to indicate its domain ! ) M Y ( t ) = & ∞ ∞-- ⋅ dy y y t e e 2 1 = & ∞-- ⋅ 2 1 dy y y t e e + & ∞- ⋅ 2 1 dy y y t e e = & ∞- ⋅ 2 1 dy y y t e e + & ∞- ⋅ 2 1 dy y y t e e = ( ) & ∞- + 1 2 1 dy t y e + ( ) & ∞- 1 2 1 dy t y e Note that the first integral converges only if t + 1 > 0, and the second integral converges only if t – 1 < 0. Therefore, the moment-generating function is only defined for – 1 < t < 1....
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Hw02ans - STAT 410 Spring 2008 Homework#2(due Friday...

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