# Calculate Expectation of points in a homogenous poission process with parameter $\alpha$ as a renewal process?

If a poisson process $N$ on $[0, \infty )$ has rate $\alpha$ (ie $E N(A)=\alpha m(A)$, $m$ lebesgue measure ) can its points be represented as occurences in a renewal process with interarrival times that are iid Exponentially distributed r.v.s with parameter $\alpha$?

I have a theorem that says that this is true for $\alpha = 1$, that the n:th point is the n:th occurence in the renewal process $\Gamma _n = E _1...+E _n$, $E _i$ iid unit exponentially distributed.

Is this true more generally, and how can this can be seen.

The reason is that I want to calculate the expectation of the third point in the homogenous poisson process $N$. Can this be done as $E [\frac {1 } {\alpha } (E _1+E _2+E _3) ]$, $E _i$ iid unit exponentially distributed?

• Yes. You might want to get hold of some textbook on Poisson processes... – Did Oct 13 '14 at 10:50
• Ok, thanks. Any recommendation? – Alexander Oct 13 '14 at 10:55
• Sure, type poisson processes lecture notes in your favorite search engine, and rejoice. – Did Oct 13 '14 at 13:37

But I will answer your actual question "calculate the expectation of the third point in the homogenous Poisson process $N$" - it doesn't require renewal theory. If $\{E_n\}$ are the arrival times then $E_n-E_{n-1}\sim\operatorname{Exp}(1)$ so $\mathbb E[E_n-E_{n-1}] = 1$ and hence $$\mathbb E[E_3] = \sum_{i=1}^3\mathbb E[E_i-E_{i-1}]= 3.$$