STAT 400 hw4_ans

# STAT 400 hw4_ans - STAT 400 Spring 2012 Homework#4(10...

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STAT 400 Spring 2012 Homework #4 (10 points) (due Friday, Feb 17, 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 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 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 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 = 1.5.
Need P( X 2 ) = ? Poisson distribution:   ! X P x x e x 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)

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## This note was uploaded on 03/14/2012 for the course STAT 400 taught by Professor Kim during the Spring '08 term at University of Illinois, Urbana Champaign.

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STAT 400 hw4_ans - STAT 400 Spring 2012 Homework#4(10...

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