Conditional homogenous Poisson Process

Question

I am currently working on a question with a homogenous poisson process. This should be simple enough in the form:

if $s < t$ and $ 0 ≤ m ≤ n $ and then calculating $P(N(s) =m|N(t) =n)$. However what should occur when $s>t$ but $n < m$ like I have in my case. At first I was assuming it would just be $0$. However I can't find any definitive answer that this is correct. I also attempted doing this with the regular rigmoral when $ 0 ≤ m ≤ n $ it but this gives me a probability > 1 so I am obviously going wrong somewhere.

Answer 1

So you have $\mathsf P(N_s=m\mid N_t=n) $ for $~0\leqslant s\leqslant t~, ~0\leqslant m\leqslant n, \;[n,m\in \Bbb N; s,t\in\Bbb R]$

You can use Bayes' Rule to "flip it around"

$$\mathsf P(N_t=n\mid N_s=m) ~=~ \dfrac{\mathsf P(N_s=m\mid N_t=n)~\mathsf P(N_t=n)}{\mathsf P(N_s=m)}$$

Or use the memoryless property of the Poisson process.

$$\mathsf P(N_t=n\mid N_s=m)=\mathsf P(N_{t-s}=n-m)$$