Suppose $X, Y, Z$ all follow a Poisson distribution each with a distinct mean parameter $\lambda_x, \lambda_y, \lambda_z$ respectively. Additionally assume distribution is independent of the others.
How would I go about calculating $P(Y < X < Z)$? My first guess would be something like
$$\sum_{k=1}^\infty \sum_{j=1}^\infty \sum_{i=1}^j f(i, j,k)$$
But I'm not sure what function to be summing over. Would it just be the product of their PMF's?
Let us use the convenient symbols $x,y,z$ as the bound variables for the series.
Due to the independence, the joint probability mass function will be the product of their probability mass functions. So the term for the series is: $$\def\pois{\operatorname{\cal Pois}}\pois(x;\lambda_{\small X})\pois(y;\lambda_{\small Y})\pois(z;\lambda_{\small Z})$$
The series needs to measure over the event of $Y{<}X{<}Z$, so the domain is: $$\{\langle x,y,z\rangle{\in}\Bbb N^3: 1{\leqslant}x, 0{\leqslant}y{\leqslant}x{-}1, x{+}1{\leqslant}z\}$$
Or such.
$$\displaystyle\mathsf P(Y{<}X{<}Z) ~=~ \sum_{x=1}^\infty\pois(x;\lambda_{\small X})\left(\sum_{y=0}^{x-1}\pois(y;\lambda_{\small Y})\right)\left(\sum_{z=x+1}^{\infty}\pois(z;\lambda_{\small Z})\right)$$