Am currently working on a Stochastic Poisson process on my project. I have thought and settled on the below scenario which I think is appropriate. However, solving it, I'm not getting what I expect.
I want to cross a road at a spot where cars pass according to a Poisson process with a rate of λ.
I will begin to cross as soon as I see there will be no cars passing for the next C time units.
I have taken N=the number of cars that pass before I cross and T= the time I begin to cross the road.
I want to determine the E(N) and Also E(T). What I know is that to find the E(T), I will have to condition on N
Let $T$ be the time the first car passes.
$ \begin{align*} E(N)&=\int_0^{\infty}E(N\mid T=t)\lambda e^{-\lambda t}\,dt \\ &=\int_0^C(1+E(N))\lambda e^{-\lambda t}\,dt \\ \end{align*} $
We solve for $E(N)$ and find that $E(N)=(1-e^{-\lambda C})e^{\lambda C}.$ Also,
$ \begin{align*} E(T)&=\int_0^{\infty}E(T\mid T=t)\lambda e^{-\lambda t}\,dt \\ &=\int_0^C(t+E(T))\lambda e^{-\lambda t}\,dt. \end{align*} $
Solving for $E(T)$ yields $\frac{e^{\lambda C}-\lambda C-1}{\lambda}$ when $\lambda,C>0$. Of course if $\lambda=0$ or $C=0$ then $E(T)=0$.