Finding the Mean and Variance of this distribution

Question

Given the following:

$Y|\Lambda=\mathrm{Poisson(\Lambda)}$ and $\Lambda=\mathrm{Gamma}(\alpha, \beta)$

Find the distribution, mean and variance of $Y$.

After a lot of work, I was able to find the distribution of $Y$ which is:

$f_Y(y)={{\Gamma(y+\alpha)e^{(-y/\beta)}}\over{y!\Gamma(\alpha)\beta^\alpha}}$

Now using this distribution, I wanted to find the mean and variance. I tried integrating the above results but I couldn't find a closed form for the product $\Pi_{i=1}^n{(y+\alpha-i)}$ and from their the integration breaks down.

I would appreciate if someone could provide me a way to calculate the mean and varaiance of Y. Thanks

Answer 1

I solved this question by setting $\beta = (p/(1-p))$. I then followed the derivation of the Negative Binomial distribution as shown in this link. Since I wanted an answer in terms of $\beta$, I simply solved the above expression for $p$ and replaced $\beta$ with that expression, giving me Y~NB($\alpha$, $\beta/(1+\beta)$). From there the mean and variance can easily be found.