Suppose $N(t)$ is a [homogeneous Poisson counting process][1] with parameter $\lambda$. Given positive real numbers $T$ and $\tau$, and non-negative integer $n$, what is the probability that $N(t)$ counts exactly $n$ points within at least one subinterval $[t,t+\tau]$ of $[0,T]$, or Prob$\big(\bigcup_t\big\{N(t)\,\big|\, [t,t+\tau]\subseteq [0,T] \wedge N(t+\tau)-N(t)=n\big\}\big)$?


  [1]: https://en.wikipedia.org/wiki/Poisson_point_process#Interpreted_as_a_counting_process