Let $N_{t}, t \geq 0$ be a Poisson process with rate $\lambda>0$ with jump times by $S_1<S_2<\cdots .$ Let $T$ be an independent geometric random variable with parameter $p \in(0,1)$, so that $\Pr(T=k)=p(1-p)^{k-1}, k=1,2, \ldots .$
Show that $S_T$ follows an exponential distribution with parameter $p \lambda$. Could I get some hints on how I can start this off?
Best Answer
\begin{align} & \Pr(S_T\le s) = \operatorname E(\Pr(S_T\le s\mid T)) \\[8pt] = {} & \sum_{t=1}^\infty \Pr(S_t\le s\mid T=t) \Pr(T=t) \\[8pt] = {} & \sum_{t=1}^\infty \int_0^s \frac 1 {\Gamma(t)} (\lambda u)^{t-1} e^{-\lambda u} (\lambda\, du) \cdot (1-p)^{t-1} p \\[8pt] = {} & \int_0^s \sum_{t=1}^\infty \frac 1 {(t-1)!}(\lambda u(1-p))^{t-1} e^{-\lambda u} (\lambda\, du)\cdot p \end{align}
This interchange of the order of summation and integration is justified by the fact that the function of $t$ and $u$ is everywhere nonnegative. That is what Tonelli's theorem tells us.
$\displaystyle \sum_{t=1}^\infty \frac{(\lambda u(1-p))^{t-1}}{(t-1)!} = e^{\lambda u(1-p)}.$