Let $X_n$ be the interarrival times for a Poisson process $\{N_t; t \geq 0\}$ with rate $\lambda$. Is it possible to calculate the probability $P\{ X_k \leq T \text{ for } k \le n, \sum_{k=1}^{n}{X_k} = t, X_{n+1}>T\}$ for given $t$ and $T$ (suppose $t$ and $T$ are compatible), i.e., how to calculate expectation of the first time that the next interarrival time is larger than or equal to T?
Thank you!