400Hw04ans - STAT 400 Fall 2011 Homework #4 (10 points)...

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STAT 400 Fall 2011 Homework #4 (10 points) (due Friday, September 23, by 3:00 p.m.) No credit will be given without supporting work. 1. Suppose a discrete random variable X has the following probability distribution: P( X = k ) = ( ) ! 2 ln k k , k = 1, 2, 3, … . Recall ( Homework #1 Problem 6 ): This is a valid probability distribution. a) Find μ X = E ( X ) by finding the sum of the infinite series. E ( X ) = x x f x all ) ( = ( ) = 1 ! 2 ln k k k k = ( ) ( ) - = 1 ! 1 2 ln k k k = ( ) ( ) ( ) - = - 1 1 ! 1 2 2 ln ln k k k = ( ) ( ) = 0 ! 2 2 ln ln k k k = 2 ln 2. b) Find the moment-generating function of X, M X ( t ). M X ( t ) = x x t x f e all ) ( = ( ) = 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 . c) Use M X ( t ) to find μ X = E ( X ). ( ) t e t e t 2 2 M ln ' X = , E ( X ) = ( ) 0 M ' X = 2 ln 2.
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d) Find σ X 2 = Var ( X ). ( ) ( ) t t e e t t e e t 2 2 2 2 M ln ln 2 X ' ' + = . E ( X 2 ) = ( ) 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 ) ) = ( ) ( ) - = 1 ! 2 1 ln k k k k k = ( ) ( ) - = 2 ! 2 1 ln k k k k k = ( ) ( ) - = 2 ! 2 2 ln k k k = ( ) ( ) ( ) - = - 2 2 2 ! 2 2 2 ln ln k k k = ( ) ( ) = 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 ) . 2. The number of typos made by a student follows Poisson distribution with the rate of 1.5 typos per page. Assume that the numbers of typos on different pages are independent. a) Find the probability that there are at most 2 typos on a page. 1 page λ = 1.5.
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Need P( X 2 ) = ? Poisson distribution: ( ) ! X P x x e x λ λ = = - P ( X 2 ) = P( X = 0 ) + P( X = 1 ) + P( X = 2 ) = ! ! ! 2 5 . 1 1 5 . 1 0 5 . 1 5 . 1 2 5 . 1 1 5 . 1 0 e e e - - - + + = 0.2231 + 0.3347 + 0.2510 = 0.8088 . b) Find the probability that there are exactly 10 typos in a 5-page paper. 5 pages
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This note was uploaded on 11/07/2011 for the course STAT 400 taught by Professor Kim during the Fall '08 term at University of Illinois, Urbana Champaign.

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400Hw04ans - STAT 400 Fall 2011 Homework #4 (10 points)...

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