Conditioning a Poisson process on the number of arrivals in a fixed time

Question

Let T1 and T5 be the first and fifth arrivals in a Poisson process with rate lambda.

(a) Find the conditional density of T1 given that there are 10 arrivals in the time interval (0,1)

(b) Find the conditional density of T5 given that there are 10 arrivals in the time interval (0,1)

I feel like this question should be really easy but i'm not sure where to get started.

Answer 1

Let $N_1$ denote the number of events in $(0,1)$. For every $n\geqslant1$, conditionally on $[N_1=n]$, the random set $\{T_k\mid1\leqslant k\leqslant n\}$ is distributed like $\{U_k\mid1\leqslant k\leqslant n\}$ where $(U_k)_{1\leqslant k\leqslant n}$ is an i.i.d. sample from the uniform distribution on $(0,1)$. This implies for example, that the event $[T_1\geqslant t]$ has the same probability as the event $[U_1\geqslant t,U_2\geqslant t,\ldots,U_n\geqslant t]$, namely, $(1-t)^n$. Thus, conditionally on $[N_1=10]$, $$ f_{T_1}(t)=10\,(1-t)^9\,\mathbf 1_{0\leqslant t\leqslant1}. $$ You might want to adapt this argument to the case of $T_5$ conditionally on $[N_1=10]$.