There are two Poisson processes with rates $\lambda_1$ and $\lambda_2$. I observe them falling into an interval of length $1$. Now, I take the arrivals for the second process and take time windows of length $w$ starting at the arrivals. Some of the arrivals from the first point process lie within these windows. Let's call this number $N$. Is it possible to get the distribution of $N$? Or say anything else about it? This distribution is a function of $\lambda_1$, $\lambda_2$ and $w$.
If the windows overlap, we just merge them into larger windows. We also chop all windows off at $1$ and ignore any events happening outside the interval. Note that if $w\geq1$, this just becomes the number of arrivals from the first process happening after the first arrival of process-2.
Some observations:
- Conditional on the total merged window size from the second process being $t$ and the number of events from the first process being $n_1$, the distribution becomes binomial. In fact, splitting a Poisson process with a coin toss results in another Poisson process. So, conditional on $t$, we have $N$ following a Poisson distribution with mean $\lambda_1 t$.
- If $w$ is small, the expected value of $N$ approaches $(\lambda_2 w) \lambda_1$. This is because the windows are so small they never overlap and total size covered by the windows has expected value $\lambda_2 w$.