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.

  • $\begingroup$ If it is $\lambda$ per unit time the answer is just $a\lambda$. $\endgroup$ – JP McCarthy Feb 10 '16 at 18:41
  • $\begingroup$ But isn't that just the expected value at time $a$? Does that take into account all the events that happened beforehand? $\endgroup$ – Ghazal Feb 10 '16 at 18:42

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

$$P[X(t+s)-X(s) = n] = P[P_o(\lambda t)=n]= e^{-\lambda t}\frac{(\lambda t)^n}{n!}$$

  • $\begingroup$ Thank you that's very helpful! $\endgroup$ – Ghazal Feb 10 '16 at 20:45
  • 3
    $\begingroup$ @Ghazal But not helpful enough to receive an upvote or an accepted answer? You are more likely to receive help in the future if you give the ppl who answer at least some credit. $\endgroup$ – JKnecht Feb 11 '16 at 0:19

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