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..
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 $\Theta$.
