I've been searching for a while but I can't seem to figure out how to find the expected value of a poisson process up to an arbitrary time.
Let {$N(t),t≥0$} be a Poisson process with rate $λ$.
How do i go about finding $E[N(t<a)]$, where "a" for the purpose of my problem is $\frac{1}{\lambda}$?
My assumption is that you have to take the expected value at each time $t$ and then sum them all, so if there were around $n$ time intervals until $t=a$, then the overall expected value would be $\sum_{t=0}^\frac{1}{\lambda} λt$ but i'm not sure if that is correct.The problem with this though, is that I don't know how many "events" occur until the time $\frac{1}{\lambda}$ so I can't necessarily compute this.
Remember that Poisson processes have stationary and independent increments.
The number of events in any time interval of length $t$ is Poisson distributed with mean $\lambda t$.
E.g. $E[N(t<a)] = \lambda a$
In the same fashion you have
$$P[X(t+s)-X(s) = n] = P[P_o(\lambda t)=n]= e^{-\lambda t}\frac{(\lambda t)^n}{n!}$$