Suppose I have two initially uncorrelated random variables, $A\sim \text{Poisson}(\mu)$ and $B \sim \text{Borel}(\lambda)$.
Suppose I now apply the constraint that $P(B > A) = 0$. That is, that random variables $B$ are prohibited to be greater than $A$.
What is the probability $P(A=i \cap B=j)$?
Attempt at a solution
Since the total probability is limited to $A<B$, I assume that one must normalise the probability to the total probability that the event can occur.
We must therefore normalise the probability to the total allowed probability, i.e. $P(B\leq A)$
\begin{equation} P(A>B) = P(B \leq A=i) = \sum^{i}_{k=0} P(B=k) \end{equation}
Therefore the conditioned probability now reads:
\begin{equation} P(A=i \cap B=j) = P(A=i)\cdot \left(\frac{P(B=j)}{\sum^{i}_{k=0} P(B=k)} \right) \end{equation}