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#1
 miodipe Member SOA Join Date: Sep 2013 Posts: 101 Need help with that problem

Hi there, I need help to solve that problem.

Suppose that the conditional distribution of N, given that Y = y, is Poisson with mean y. Further suppose that Y is a gamma random variable with parameters (r, λ), where r is a positive integer.

(a) Find E[N].
(b) Find Var(N).
(c) Find P(N = n)

My work so far.
a) E(N)=E(E(N|Y)=E(Y)=rλ
b)Var(N)=E(Var(N|Y))+Var(E(N|Y))=E(Y)+Var(Y)=rλ+rλ ^2

Is my work correct so far, and also I am not sure how to start part c.

Thanks for your help!
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#2
 C2H6O Member Non-Actuary Join Date: Dec 2013 Posts: 56 Part (a) and (b) are correct.

For finding P(N = n), multiply the PMF of N|Y with the PDF of Y and integrate over the range of Y.
#3 AndyA Member SOA Join Date: Aug 2013 College: Florida Atlantic University Alumni Posts: 660 Quote:
 Originally Posted by miodipe Hi there, I need help to solve that problem. Suppose that the conditional distribution of N, given that Y = y, is Poisson with mean y. Further suppose that Y is a gamma random variable with parameters (r, λ), where r is a positive integer. (a) Find E[N]. (b) Find Var(N). (c) Find P(N = n) My work so far. a) E(N)=E(E(N|Y)=E(Y)=rλ b)Var(N)=E(Var(N|Y))+Var(E(N|Y))=E(Y)+Var(Y)=rλ+rλ ^2 Is my work correct so far, and also I am not sure how to start part c. Thanks for your help!

If N is Poisson and its parameter is a Gamma random variable then the posterior predictive distribution is a Negative Binomial distribution. I believe the parameters of the Negative Binomial distribution are the same as those of the Gamma conjugate prior distribution. If you use this trick you won't need to integrate the gamma pdf.
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#4
 miodipe Member SOA Join Date: Sep 2013 Posts: 101 Thanks for your help!
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