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Let $(T_n)_{n \geq 0} $ be a sequence of random variables such that $T_0 =0$ and $(T_n -T_{n-1})_{n \geq 1}$ are independent exponential random variables with parameter $\theta >0$. Can someone help me to find the joint distribution (i.e. a joint probability mass function) of $$ \bigg( \sum_{n \geq 1} \mathbf{1}_{ \{ T_n \leq t \} } \bigg)_{t \geq 0} \text{ at finite time points } t_1 < t_2 < \ldots < t_n.$$

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Ok, so apparently you're aware that each $N_t$ is a Poisson variable and that exponential variables have no memory.

Use the no-memory property to prove that $N_{t_2}-N_{t_1}$ is independent of $N_{t_1}$ (actually the whole process $N_s-N_{t_1}, s>t_1$, is independedent of $N_u,u<t_1$).

Then you can easily see that $N_{t_2}-N_{t_1}$ is a Poisson variable with parameter $t_2-t_1$ (because it is exactly the same oas proving that $N_t$ is Poisson with parameter $t$).

Final step: your distribution is that of $(N_1,N_1+N_2,...,N_1+...+N_q)$ where these variables are independent and $N_i$ is Poisson with parameter $t_i-t_{i-1}$ (with $t_0=0$)

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  • $\begingroup$ I know this, but I am looking for the joint distribution at times $t_1 < t_2 < \ldots < t_n$. $\endgroup$ – erik Jan 3 '15 at 4:34

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