TITLE: Find the probability distribution function
QUESTION [1 upvotes]: Let $X$ and $Y$ be random variables such that $Y|X = x$ is $\text{Exp}(1/x)$ and X is $\text{Gamma}(2,1)$ find the pdf of $Y$ and $E(Y)$

REPLY [0 votes]: Hint:
$f_{Y|X}(y|x)f_X(x)=f(x,y)\\
f_Y(y)=\int_{-\infty}^{\infty} f(x,y)dx$
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