Let $X,Y,Z$ be independent discrete random variables. Calculate $P(X<Y<Z)$
Solution 1:
For your Poisson example (or any discrete one):
$$ P(X<Y<Z) = \sum_{x<y<z}P(X=x,Y=y,Z=z) = \sum_{x=0}^{\infty}\sum_{y=x+1}^{\infty}\sum_{z=y+1}^{\infty}P(X=x,Y=y,Z=z) $$
If you are blessed with independence, then $P(X=x,Y=y,Z=z)$ is easy to calculate at least in the Poisson case. And those scary infinity summations are gonna be easy to deal with by the "everything sum to one" property of Poisson distribution (every distribution indeed), you just have to be careful with the index.