I'm having some trouble with something my professor said would be on our exam. For a Poisson process $N_t$ with interarrival times $X_i$, how is it that you find $E(X_i | N_t = n)$ (assuming $n\ge i$)? Similarly having trouble with finding $Var(X_i | N_t = n)$, but if I had one I could probably figure out the other pretty easily.
I'm thinking the solution might have something to do with the joint pdf of the arrival times, but I also might be overthinking it a bit. Any help with how to at least start this would be appreciated!
The key result behind the answer is the following theorem:
If $N$ is a homogeneous Poisson process over $[0,\infty)$ with some finite rate $k$, then the $n$ "events" of the Poisson process $X_i$, in increasing order and based conditionally on the fact that $n$ events occured, are distributed as the order statistics from a sample of size $n$ from the uniform distribution $U(0,t)$ on $[0,t]$.
Less formally, it means that I can take $n$ samples from a uniform distribution defined between $0$ and $t$, place them in order on that time interval, and they would be probabilistically identical to the interarrival points of a Poisson process over that same interval as long as only precisely $n$ arrival events occurred.
Two proofs with different levels of detail are discussed in Resnick's book and in many other stochastic processes textbooks. Hope this helps!