I have stumbled upon an interesting problem with Poisson-distributed random variables the other day and I wasn't really able to come up with a solution - I wonder if anyone could help me out here. The question is as follows. Suppose you have two i.i.d. random variables $X_1$ and $X_2$ s.t. $X_1 \sim Poisson(\lambda)$ and $X_2 \sim Poisson(\lambda)$. Now let's draw a large number of pairs $(x_1, x_2)$ from $(X_1, X_2)$ and for each pair, do the following:
- If $x_1$ > $x_2$, store $x_1$ in a list $L$.
- If $x_1$ < $x_2$, do nothing, take the next pair.
- If $x_1$ = $x_2$, store $x_1$ in $L$ with probability 0.5.
The question is: what is the distribution of numbers that we have placed in $L$?
Note: I made a few numeric simulations and for me the distribution of numbers in $L$ looks like another Poisson distribution with a mean higher than $\lambda$, but I can neither prove nor disprove it. Is there a closed-form solution for this? If not, is it possible to work out at least some kind of an iterative equation that I can use to obtain the distribution for any given $\lambda$?