Let $X$ and $Y$ be random variables such that $X \vert Y=y$ is normal distributed as $N(y,1)$ and Y is a continues random variable with PDF:
and $0$ otherwise.
My idea is to find the marginal PDF for X and from there it's easy to find the variance. Now since the conditional is normal distributed, I should be able to find the simulatanous PDF by multiplying the the conditional PDF by the marginal PDF for Y. However when I try to use integration to find the marginal PDF for X with limits 0 and infinity, I get an uncomputable integral.... Why is this happening and/or is there another easier method? Thanks in advance!