If one has multiple Poisson random variables (with potentially different lambda parameter), how can one theoretically find the probability that one of those variables is greater than the others?
That is, assume you have $k$ random variables $X_{i=1...k}$ s.t. $X_i \sim \text{Pois}(\lambda_i)$, where lambdas are not necessarily the same. I want to find $P(X_i > X_j) \ \forall i \neq j : 1 \leq j \leq k$.
I am aware that this may not have a closed form. In fact, I think it is just a generalization of what can be seen in this question. However, I am struggling with how to write down such generalization: at first, I thought that it would be just a product of the calculation of the formula in the above mentioned question, for each pairwise comparison. However, this is wrong, since the pairwise comparisons are not independent events. That is, $P(X_i > X_j) $ is not independent from $P(X_i > X_h) \ \forall h \neq i,j : 1\leq h \leq k$.
What is the correct formula for finding the probability that the realization of a given Poisson random variable will will be greater than the realizations of $k-1$ other random variables?
For $3$ random variables for example, the probability that $X_1$ is greater than both $X_2$ and $X_3$ is $$\sum_{k=0}^\infty \Pr(X_1= k+1)\Pr(X_2\leq k)\Pr(X_3\leq k)$$ assuming the random variables are independent. Then fill in the formulas for these probabilities, just as in the question you linked. The generalization to an arbitrary number of random variables should be clear.
This will be difficult to evaluate, I should think. You might have better luck with simulation.