I dont' really know how to realize the following: Let $\lambda \sim \Gamma(\alpha,\beta)$ and let $X$ conditional on $\lambda$ be Poisson$(\lambda)$. Argue that for $n=0,1,2,\ldots,$ $P(X=n)=\int_0^\infty \frac{\lambda^n}{n!}e^{-\lambda}\frac{\beta^\alpha}{\Gamma(\alpha)}\lambda^{\alpha-1}e^{-\beta\lambda} \, d\lambda.$ I don't know how to deal with this type of conditional probability. I'd appreciate some help.
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