Conditioning on arrival time in poisson process

Question

Suppose there is a poisson process with some parameter $\lambda$. Suppose $n_t$ is the $t^{th}$ arrival time. Let $N(t) = \max\{n: {t_n} {\leq} t\}$. Is the following true: $\Pr(N(t)=2) = \Pr(t_2 = t)$?

I was confused about this when I was trying to calculate $\Pr(t_1 {\leq} s {\vert} t_2 = T)$.

Thanks!

Answer 1

No.

Since a Poisson process is continuous, $t_2$ is a continuous random variable meaning you have $\Pr(t_2 = t) =0$ but $\Pr(N(t)=2) \gt 0$ at least for $t \gt 0$

You can say $\Pr(N(t)=2) = \Pr(t_2 \le t \text{ and } t_3 \gt t)$