If X~Po(2) and Y~Po(1). How to find P(X>Y)?
The joint pdf is $\displaystyle f(k, l) = \left( \frac{e^{-2} 2^k}{k!} \right) \, \left( \frac{e^{-1} 1^l}{l!} \right) = \frac{e^{-3} 2^k}{k! \, l!}$.
$\displaystyle \Pr(X > Y) = \sum_{k=1}^{\infty} f(k, 0) + \sum_{k=2}^{\infty} f(k, 1) + \sum_{k=3}^{\infty} f(k, 2) + \, .... $