If we have a Poisson-process with rate $\lambda$, the time it takes for $k$ events to happen is the sum of $k$ exponentially distributed RVs.
Now consider some interval $[x, x+\hat{t}$]. Let $T_i$ be the time the i-th event (in that interval) occurs and $N$ the number of events in that interval.
How to find $Pr(T_k < t | N \geq k)$?
What I did so far:
Calculated the number of events:
$i\geq k: Pr(N = i | i \geq k) = \frac{P_\lambda(k)}{\sum_{i=k}^{\infty} P_\lambda(i)} = \frac{P_\lambda(k)}{1-\sum_{i=0}^{k-1} P_\lambda(k)}$ where $P_\lambda$ is the Poisson-distribution $i < k: Pr(N = i | i \geq k) = 0$
I tried to find $Pr(T_k < t | N \geq k)$ but I don't know how to properly include the time upper-limit $\hat{t}$. I also tried finding a recursive relationship, calculating $Pr(T_k < t | N \geq k)$ using $Pr(T_{k-1} < t | N \geq k-1)$.