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Short-Term Actuarial Math Old Exam C Forum

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Old 01-25-2015, 07:26 PM
miodipe miodipe is offline
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Default 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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Old 01-25-2015, 08:06 PM
C2H6O C2H6O is offline
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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.
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Old 01-26-2015, 02:46 AM
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AndyA AndyA is offline
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Quote:
Originally Posted by miodipe View Post
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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Old 01-26-2015, 08:36 PM
miodipe miodipe is offline
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Thanks for your help!
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