I lack the basic probability knowledge to describe my problem with all the correct keywords, so I will try it in the form of a puzzle:
There are two guys, $A$ and $B$, each with a button they can press. The number of times that a guy presses a button in an interval of time follows a Poisson distribution with means $\lambda_A$ and $\lambda_B$ respectively. Every time a guy pushes the button, they generate a random integer $x \in [0, X]$ with probabilities $P_A(x)$ and $P_B(x)$ respectively. The interval of integers is the same for both, but the distribution with which the integers are generated is different for both guys.
Their buttons are connected to a single screen, so I can only see one integer at a time. My task is, given an integer in my screen (an event), to select events generated by $B$ and let pass events generated by $A$ with the highest accuracy possible.
Does anybody have any suggestions on how to tackle this?
So far I am trying to first deal with the simplest case, where both $A$ and $B$ press buttons at the same rate. I know the probability distributions of the integers that $A$ and $B$ generate. In this case, does it make sense to just classify events as $A$ or $B$ based on who's probability distribution gives a higher value for a given integer?
I appreciate any help or reading material in the correct direction.