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Unformatted text preview: SOLUTION FOR HOMEWORK 4, STAT 4382 Well, welcome to your fourth homework. Exam 1 is based on all 4 homeworks. 1. Problem 2.1.4. Here X is Binomial ( p, N ) given N which is a Binomial ( q, m ) random variable. We are asked to find the mean of X . Write, E ( X ) = E ( E ( X  N )) = E ( pN ) = pE ( N ) = pqm. I skip plugging numbers. 2. Problem 2.1.9. Here N is Poisson ( λ ), and given N the random variable X is discrete uniform on the set { , 1 , . . . , N + 1 } . We are asked to find the marginal distribution of X . Well, we begin with the joint distribution p X,N ( x, n ) = p N ( n ) p X  N ( x  n ) = [ e λ λ n /n !][1 / ( n + 2)] I ( x ∈ { , 1 , . . . , n + 1 } ) . Now we can find the marginal (pay attention to the fact that given x the random N cannot be smaller than x 1 p X ( x ) = ∞ summationdisplay n =min(0 ,x 1) e λ λ n / [ n !( n + 2)] = ∞ summationdisplay n =min(0 ,x 1) e λ λ n [ 1 ( n + 1)! 1 ( n + 2)! ] . I stop here and will allow you to finish. Note that we arrived at difference of two terms where each term can be calculated using the fact that a Poisson probability mass function is added to 1....
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This note was uploaded on 02/23/2012 for the course STAT 4382 taught by Professor Krwaw during the Fall '11 term at King Mongkut's Institute of Technology Ladkrabang.
 Fall '11
 krwaw
 Binomial

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