Let $ Y \sim Gamma(\alpha, \lambda) $ and $X|Y= y\sim \text{POI}(y)$. Find the conditional distribution $Y|X=x$
How to do that? I am confused because the Gamma distribution is a continous distribution and Poisson distribution is a discrete distribution. Can it be solved?
Thanks for any help
The knowledge of the distributions of $Y$ and $X|Y=y$ gives the joint distribution of $(X,Y)$ from which the conditional distribution of $Y|X=x$ can be easily computed.
Indeed the density of $(X,Y)$ w.r.t the product measure $N\otimes \lambda$ where $N$ is the counting measure on $\mathbb N$ is $$f_{X,Y}(k,y) \propto y^{\alpha+k-1}e^{-(\beta+1)y}$$ (removing all the factors that do not depend on $y$)
so $Y|X=k \sim Gamma(\alpha+k, \beta+1) $