How to derive the marginal probability function of X?

Question

Let $X$ and $Y$ be discrete random variables with joint probability function $f(x,y)=k\frac{2^(x+y)}{x!y!}$ for $x=0,1,2..$ and $y=0,1,2...$,where $k$ is a positive constant. The answer is $k\frac{(2^x)(e^2)}{x!}$. I do not know how to get the $e^2$. How to derive the marginal probability function of $X$? Please help. Thank you very much!

Answer 1

We find the (marginal) distribution function of $X$, in a harder than necessary way. We want $\Pr(X=x)$. This is $$\sum_{y=0}^\infty f(x,y).$$ So we want $$k\frac{2^x}{x!}\sum_{y=0}^\infty \frac{2^y}{y!}.$$ We recognize the inner sum as the power series expansion of $e^2$.

Note that if we sum over all $x$, we now get in the same way $ke^4$. Thus $k=e^{-4}$.