I want to know how to compute joint pdf of (x,y) where x follows exponential distribution with parameter $\lambda$ and $y=x+\epsilon $, $\epsilon$ follows standard normal distribution. x and $\epsilon$ are independent.
and how to compute pdf of y.
the question I met is to show $f(x|y)=\frac{\phi(x-(y-\lambda))}{\Phi(y-\lambda)}$, where $\phi,\Phi$ are pdf and cdf of standard normal. I guess this equation is originated from $f(x|y) = \frac{f(x,y)}{f(y)}$, so I need to compute the joint pdf and the pdf of y. I tried but can't get the same form as shown above.