# Find the joint probability function of X and Y.

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.

• You can proceed very much for this mixed joint distribution (en.wikipedia.org/wiki/Joint_probability_distribution#Mixed_case) as you would for discrete or continuous cases, starting from $f_{X,Y}(x,y)=f_{Y|X}(y|x)f_X(x)$. – Mick A Dec 13 '15 at 1:54
• So i guess its a simple solution in which I should bring $f(x)$ and $f(Y|X=x)$ together through simple multiplication. I get $\frac{x^y}{y!}e^{-2x}$ – amundi32 Dec 13 '15 at 3:40
• Yes, that's right. – Mick A Dec 13 '15 at 7:37