Consider two independent Poisson process A, B which have rates $\lambda_A$ and $\lambda_B$. The question is to find the distribution of difference of time between event B and the last event A before it. At first sight I thought that since we can reverse time in Poisson process, so the last A before B is simply the first A after B, then the distribution should be
$$f(t)=\lambda_A e^{-\lambda_A t}$$
But I did some Monte Carlo simulation and the result turned out to be
$$f(t)=(\lambda_A+\lambda_B)e^{-(\lambda_A+\lambda_B)t}$$
What am I missing here?
