Problem. Let $(M_t)_{t\geq 0}$ and $(N_t)_{t\geq 0}$ be Poisson processes with intensities $\alpha > 0$ and $\beta > 0$, respectively, on a probability space $(\Omega, \mathcal{F}, \mathbb{P}$). Let $(S_k)_{k\in\mathbb{Z}_+}$ and $(T_k)_{k\in\mathbb{Z}_+}$ be the jump processes for $(M_t)_{t\geq 0}$ and $(N_t)_{t\geq 0}$, respectively. Suppose that $S_k$ and $T_l$ are independent for all $k, l \in \mathbb{Z}_+$.
(a) Define the random variable $N_{S_1} : \Omega \to \mathbb{Z}_+$ by $N_{S_1}(\omega) = N_{S_1(\omega)}(\omega)$ for $\omega \in \Omega$. What is the probability distribution of $N_{S_1}$? Is it a standard distribution that we have encountered before? Hint: first write $\mathbb{P}(N_{S_1} \geq n)$ as a double integral.
I was hoping to get some help with this question. I'm just really stuck on how to write this probability as a double integral.
Any help would be appreciated.
Thanks
By the independence, we know that $(N_t)$ and $(S_k)$ are independent. Then
\begin{align*} \mathbb{P}(N_{S_1} \geq n) = \mathbb{P}(T_n \leq S_1) &= \int_{0}^{\infty} \int_{0}^{\infty} \mathbf{1}_{\{t \leq s\}} f_{S_1}(s)f_{T_n}(t) \, dsdt \\ &= \int_{0}^{\infty} \left( \int_{t}^{\infty} \beta e^{-\beta s} \, ds \right) \frac{\alpha^n t^{n-1} e^{-\alpha t}}{(n-1)!} \, dt \\ &= \int_{0}^{\infty} \frac{\alpha^n t^{n-1} e^{-(\alpha+\beta) t}}{(n-1)!} \, dt \\ &= \left( \frac{\alpha}{\alpha+\beta} \right)^n. \end{align*}