I have the following question in my homework:
$N(t)$ is a renewal process where $L(N(t) + 1) = \operatorname{Geometric}(\exp(-t))$. Find $E[T_1]$ and $E[T_2]$.
However, I'm not sure where to start in order to solve this question. I do know that:
$\lbrace N(t) = k-1\rbrace = \lbrace T_{k-1} \leq t\rbrace \cap \lbrace T_{k} > t \rbrace$
Me and my teammate have already attempted something for $E[T_1]$, however, I don't use the fact that $L(N(t) + 1) = \operatorname{Geometric}(\exp(-t))$, so I'm not entirely convinced of my answer. My answer is the following:
$P[T_1 > t] = P[N(t) = 0] = e^{-t}$
$P[T_1 < t] = F_{T_1}(t) = 1 - e^{-t}$
$f_{T_1} = \dfrac{d}{dt}F_{T_1}(t) = e^{-t}.$
Therefore, $T_1$ follows an exponential distribution with parameter $\lambda = 1$ and $E[T_1] = 1/\lambda = 1$.
I still haven't managed to figure something out for $E[T_2]$.
