Hw06ans - STAT 408 Spring 2009 Homework#6(due Friday March...

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STAT 408 Spring 2009 Homework #6 (due Friday, March 6, by 3:00 p.m.) 1. Suppose a discrete random variable X has the following probability distribution: P( X = k ) = ( 29 ! 2 ln k k , k = 1, 2, 3, … . a) Verify that this is a valid probability distribution. f ( x ) 0 2200 x c ( 29 x x f all = 1 ( 29 = 1 ! 2 ln k k k = ( 29 = 0 ! 2 ln k k k – 1 = e ln 2 – 1 = 2 – 1 = 1. c b) Find μ X = E ( X ) by finding the sum of the infinite series. E ( X ) = x x f x all ) ( = ( 29 = 1 ! 2 ln k k k k = ( 29 ( 29 - = 1 ! 1 2 ln k k k = ( 29 ( 29 ( 29 - = - 1 1 ! 1 2 2 ln ln k k k = ( 29 ( 29 = 0 ! 2 2 ln ln k k k = 2 ln 2. c) Find the moment-generating function of X, M X ( t ). M X ( t ) = x x t x f e all ) ( = ( 29 = 1 ! 2 ln k k k t k e = = 1 ! 2 ln k k t k e = 1 2 ln - t e e = 1 2 - t e . d) Use M X ( t ) to find μ X = E ( X ). ( 29 t e t e t 2 2 M ln ' X = , E ( X ) = ( 29 0 M ' X = 2 ln 2.
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e) Find σ X 2 = Var ( X ). ( 29 ( 29 t t e e t t e e t 2 2 2 2 M ln ln 2 X ' ' + = . E ( X 2 ) = ( 29 0 M ' ' X = 2 ( ln 2 ) 2 + 2 ln 2. Var ( X ) = E ( X 2 ) [ E ( X ) ] 2 = 2 ln 2 – 2 ( ln 2 ) 2 = 2 ln 2 ( 1 – ln 2 ) . OR E ( X ( X – 1 ) ) = ( 29 ( 29 - = 1 ! 2 1 ln k k k k k = ( 29 ( 29 - = 2 ! 2 1 ln k k k k k = ( 29 ( 29 - = 2 ! 2 2 ln k k k = ( 29 ( 29 ( 29 - = - 2 2 2 ! 2 2 2 ln ln k k k = ( 29 ( 29 = 0 2 ! 2 2 ln ln n n n = 2 ( ln 2 ) 2 . E ( X 2 ) = E ( X ( X – 1 ) ) + E ( X ) = 2 ( ln 2 ) 2 + 2 ln 2. Var ( X ) = E ( X 2 ) [ E ( X ) ] 2 = 2 ln 2 – 2 ( ln 2 ) 2 = 2 ln 2 ( 1 – ln 2 ) .
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2. Let a > 2. Suppose a discrete random variable X has the following probability distribution: p ( 0 ) = P ( X = 0 ) = c , p ( k ) = P ( X = k ) = k a 1 , k = 1, 2, 3, … . a)
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This note was uploaded on 04/27/2009 for the course STAT 420 taught by Professor Stepanov during the Spring '08 term at University of Illinois at Urbana–Champaign.

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Hw06ans - STAT 408 Spring 2009 Homework#6(due Friday March...

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