Question
Suppose X ~ Pois($\lambda$), where the mean parameter $\lambda$ itself is a random variable with pdf
$$g(\lambda;\alpha,\beta) = \frac{\beta ^\alpha}{\Gamma(\alpha)} \lambda^{\alpha - 1} \exp(-\beta \lambda)$$
and $\lambda, \alpha, \beta > 0$
Find the distribution of X and compute $\mathbb{E}(X)$.
My working
$$\begin{aligned} \mathbb{P}(X=x) & = \int_{0}^{\infty} \mathbb{P}(X = x \mid \lambda = \lambda) \mathbb{P}(\lambda = \lambda) \ d\lambda \\ & = \int_{0}^{\infty} \lambda e^{-\lambda x}\frac{\beta^{\alpha}}{\Gamma(\alpha)}\lambda^{\alpha - 1} e^{-\beta \lambda} \ d\lambda \\ & = \frac{\beta^{\alpha}}{(x+\beta)^{\alpha -1}} \end{aligned}$$
Now I am stuck as I am not sure if my answer for distribution of X is correct. If I am on the right path, how should I proceed to compute $\mathbb{E}(X)$?
Any intuitive explanations will be greatly appreciated! :)