For a Poisson process with rate $\lambda$. Compute the conditional probability, $P(X(s) = x | X(t)=n)$, given $s > t$.

Question

I saw a post discussing this question, but the condition given is that $0<s<t$. Let $\{X(t)\}$ be a Poisson process with arrival rate $\lambda>0$. Compute the conditional probability, $P(X(s) = x|X(t) = n)$.

If the condition is changed to $s > t$, will it make any difference?

I would think that having $s>t$ makes the two events independent, but I am not sure. Can anyone explain this to me?

Answer 1

You are correct about independence: $x-n$ events occurring after time $t$ has nothing to do with what happened before $t$. So:$$ P(X(s) = x | X(t)=n)=P(X(s-t)=x-n) $$