I observe a Poisson process for a finite time interval. Let $X_i$ denote the time of the $i^{th}$ arrival. I start my timer as soon as I get my first event so that $X_0:=0$ and store the time-stamps $0=:X_0<X_1<X_2<\ldots <X_{N(T)}<T$ for a random number of arrivals $N(T)$ in the time interval $[0,T)$.
Let $Y_k = X_{k}-X_{k-1}$, $1\leq k \leq N(T)$, be the sequence of interarrival times. How do I find the distribution of the mean interarrival time: $m(T):=\frac{1}{N(T)}\sum_{i=1}^{N(T)} Y_i = \frac{X_{N(T)}}{N(T)}$?
(If $N(T)=0$, I define $m(T)=\infty.$)