I'm having trouble determining the conditional probability of the sum of two independent Poisson Point Processes, $X, Y$, with parameters $\lambda,\mu$ respectively. If $X+Y=W$, I would like to find:
$$P(W_t=w|Y_s=y)$$
where $s<t$. I tried using Bayes' Rule because I thought the following form might be simpler to solve:
$$\rightarrow P(X_t+Y_t=w|Y_s=y)=\frac{P(Y_s=y|X_t+Y_t=w)\cdot P(X_t+Y_t=w)}{P(Y_s=y)}$$
The right term in the numerator and the term in the denominator are simple enough to calculate. The left term in the numerator is:
$$\rightarrow \frac{P(Y_s=y, X_t+Y_t=w)}{P(X_t+Y_t=w)}$$
The denominator here cancels with the right term in our original numerator, so we have:
$$\rightarrow \frac{P(Y_s=y, X_t+Y_t=w)}{P(Y_s=y)}$$
It is here that I am stuck, as I don't know how to separate the numerator here into two separate, calculatable probabilities. I was thinking I could do some simplification like this:
$$\rightarrow \frac{P(Y_s=y, X_t=w-y)}{P(Y_s=y)}=\frac{P(Y_s=y)P(X_t=w-y)}{P(Y_s=y)}=P(X_t=w-y)$$
But I'm not sure how, if at all, this is justifiable. Any suggestions would be sincerely appreciated. Cheers.