# What is the unconditional distribution?

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.

• 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. – Nick Peterson Oct 17 '15 at 16:34

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 $\Theta$.