I am trying to find the variance of $X$ which is defined like this:
$$X \sim N(Y,e)$$
where $Y$ is a normal random variable with the distribution $Y \sim N(a,b)$. $a$,$b$, and $e$ are known constants.
How can I go about doing this? I set up an integral like this to get the pdf of $X$:
$$\int_{-\infty}^{\infty}f_{X}(x)f_{Y}(y)\text{ d}y$$
but have no idea how to solve the integral. please help!
Answered my question!
The conditional variance formula:
Var(X) = E[Var(X|Y)] + Var(E[X|Y]) = e + b.