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, = 6, = 0.20. E ( X ) = = 1.2, Var ( X ) = 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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