The number of defects per yard $Y$ for a certain fabric is known to have a Poisson distribution with parameter $x$, i.e. $f(Y|X = x)$ has a Poisson distribution with parameter $x$. However, $X$ itself is a random variable with probability density function given by:
$$f(x) = e^{-x}, \quad \text{ for } x \geq 0$$
Find the joint probability function for $X$ and $Y$.
It confuses me a bit since $Y$ is a discrete random variable whereas $X$ is continuous as given by the variable. Apologies for the lack of attempt since I don't know where to begin in this case.