I am wondering how the waiting times are distributed for the poisson process, conditioned on a number of events by time t. Look at this theorem:
Here, the S's are the sum of the waiting time to event n. But I am interested in the $T_i's$. I know that unconditionally these are exponentially distributed, but what will the conditional distribution be? I tried to calculate these but I got $\frac{n!}{t^n}$ in this case also(the joint distribution of the T's), but this can't be?, since the S's and T's obviously can't have the same distribution?
The conditional distribution of $(S_k)_{1\leqslant k\leqslant n}$ is uniform, that is, its density is $$n!\,t^{-n}\,\mathbf 1_D,$$ where $$ D=\{(s_k)_{1\leqslant k\leqslant n}\mid 0\leqslant s_1\leqslant\cdots\leqslant s_n\leqslant t\}. $$ One is interested in the conditional distribution of $(T_k)_{1\leqslant k\leqslant n}$ where $T_1=S_1$ and $T_{k+1}=S_{k+1}-S_k$ for every $k\geqslant1$. A change of variable does the job.
To wit, if $t_1=s_1$ and $t_{k+1}=s_{k+1}-s_k$ for every $k\geqslant1$, then the domain $D$ becomes $$ \Delta=\{(t_k)_{1\leqslant k\leqslant n}\mid \forall k,\,t_k\geqslant0,\ t_1+\cdots+t_n\leqslant t\}, $$ and the Jacobian is $1$, hence the conditional distribution of $(T_k)_{1\leqslant k\leqslant n}$ is uniform, that is, its density is $$n!\,t^{-n}\,\mathbf 1_\Delta.$$