Conditional distribution on arrival time (Poisson process)

Suppose that $\{N_t: t\geq 0\}$ is a Poisson process of rate $\lambda$ and $T_1< T_2< \dotsb\$ are its arrival times (i.e. $T_i := \min \{t\geq 0 : N_t \geq i\}$). What is the conditional distribution of $N_2$, given $T_3=1$?

Bonus: I am wondering whether it is possible to generalize this to something like:

Let $0<t_0<t_1$ and let $n$ be a positive integer. What is the conditional distribution of $N_{t_1}$, given $T_n=t_0$?

• I tried rewriting $T_3=1$ as $\{N[0,1)=2, N[1,1+\epsilon)=1\}$ for suitably small $\epsilon$ and using the fact that $N[1+\epsilon,2)$ is independent of $N[0,1)$ and $N[1,1+\epsilon)=1$, letting then $\epsilon$ tend to 0, but I am not sure if it is correct. – AdrianoMeis Mar 31 '15 at 17:04
• $P(N_2=n|T_3=1)$ means that the third jump was at $t=1$, so you're asking what is the probability of $N$ in $t=2$, given $N_1=3$. Therefore: $P(N_2=n|T_3=1)=P(N_2=n|N_1=3)=P(N_2-N_1=n-3|N_1=3)$ and by the increment independence, this is $P(N_2-N_1=n-3)$, and this is Poisson distribution with parameter $\lambda$ and argument $n-3$. – yoki Mar 31 '15 at 17:05
• @yoki This is what I obtained as a result proceeding as in the comment above, but I wasn't sure it was correct. Thank you! – AdrianoMeis Mar 31 '15 at 17:06
• Look up results on connections between Poisson, exponential, and order statistics of uniform. – BruceET Mar 31 '15 at 17:54
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