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Suppose that Θ is a random variable that follows a gamma distribution with parameters λ and α, where α is an integer, and suppose that, conditional on Θ, X follows a Poisson distribution with parameter Θ. Find the unconditional distribution of α + X (Hint : Find the mgf by using iterated conditional expectations.

please answer..

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  • $\begingroup$ How do you define this particular gamma distribution? There are a couple of different parameterizations of it, and I don't generally see them with the parameters being called $\lambda$ and $\alpha$; so, I'm not sure which one you're getting at. $\endgroup$ – Nick Peterson Oct 17 '15 at 16:34
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Hint: You're asked to consider the moment generating function of $\alpha+X$, which is $$ M_{\alpha+X}(t):=\mathbb{E}[e^{t(\alpha+X)}]. $$ Because $\alpha$ is constant, we can write $$ \mathbb{E}[e^{t(\alpha+X)}]=\mathbb{E}[e^{t\alpha}\cdot e^{tX}]=e^{t\alpha}\mathbb{E}[e^{tX}]=e^{t\alpha}M_X(t). $$ So, we really just need to compute $M_X(t)$ to finish the problem.

Now, we know that conditioned on $\Theta$, $X\sim\text{Poisson}(\Theta)$; using this, you can show fairly easily that $$ \mathbb{E}[e^{tX}\mid\Theta]=\exp[\Theta(e^t-1)]. $$ However, we know that $$ \mathbb{E}[e^{tX}]=\mathbb{E}[\mathbb{E}[E^{tX}\mid\Theta]], $$ so that $$ M_X(t)=\mathbb{E}[\exp[\Theta(e^t-1)]] $$ where this last expectation is taken with respect to $\Theta$.

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  • $\begingroup$ Thanks. then, unconditional distribution of α + X is exp(tα)expectation(exp(Θ(exp(t)-1))) ? I'm not understanding mean of unconditional dist. unconditional dist is equal to m.g.f ? $\endgroup$ – Sung Pang Oct 17 '15 at 17:07
  • $\begingroup$ No; that is it's moment generating function, though. $\endgroup$ – Nick Peterson Oct 17 '15 at 17:09
  • $\begingroup$ Please tell me technique (distribution from moment generating function.) $\endgroup$ – Sung Pang Oct 17 '15 at 17:15
  • $\begingroup$ In this sort of problem: you should find that the moment generating function corresponds to a known (common) probability distribution. $\endgroup$ – Nick Peterson Oct 17 '15 at 17:16
  • $\begingroup$ Hmm.. I'm learning basic mathematical statistics.. very very difficult. I'll try it !! Thanks :) $\endgroup$ – Sung Pang Oct 17 '15 at 17:25

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