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solhw43824 - SOLUTION FOR HOMEWORK 4 STAT 4382 Well welcome...

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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 { 0 , 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 ∈ { 0 , 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. 3. Problem 2.3.2. Here N is Binomial ( p,m ) with p = 1 / 2 (fair coin) and m = 6. Then, given N , Z is Binomial ( q,N ) with q = 1 / 2. First, we are asked to find the mean and
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