Finding joint distribution when one variable is a parameter of another.

Question

Let $U \sim U(1, 2)$ and $X \sim Exp(U)$. How would I find the joint distribution of $X$ and $U$ in this case?

Answer 1

You should use relation between conditional density and joint density. $f_{(X,U)}(x,u)=f(x|u)f(u)=\frac{1}{u}e^{-\frac{x}{u}}1_{(0,\infty)}(x) f(u)= \frac{1}{u}e^{-\frac{x}{u}} 1_{(0,\infty)}(x) 1_{(1,2)}(u)$