There is a poisson process N with rate $\lambda$. $T_1$ and $T_2$ are the first and second arrival times.
How do you calculate $P(T_1 \le t_1 , T_2 \le t_2 , N_t=2) $
There is a poisson process N with rate $\lambda$. $T_1$ and $T_2$ are the first and second arrival times.
How do you calculate $P(T_1 \le t_1 , T_2 \le t_2 , N_t=2) $
Conditionally on $[N_t=2]$, the set $\{T_1,T_2\}$ is distributed as $\{U,V\}$ with $U$ and $V$ i.i.d. and uniform on $[0,t]$. In particular, for every $0\lt s\lt r\lt t$, $ \mathbb P(s\lt T_1,T_2\lt r\mid N_t=2)=\mathbb P(U\in[s,r],V\in[s,r])=t^{-2}(r-s)^2. $ Thus, $ \mathbb P(s\lt T_1,r\lt T_2\mid N_t=2)=t^{-2}((t-s)^2-(r-s)^2). $ Finally, $N_t$ is Poisson with parameter $\lambda t$ hence $\mathbb P(N_t=2)=\tfrac12\lambda^2t^2\mathrm e^{-\lambda t}$ and $ \mathbb P(s\lt T_1,r\lt T_2,N_t=2)=\tfrac12\lambda^2\mathrm e^{-\lambda t}((t-s)^2-(r-s)^2). $