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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?

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  • $\begingroup$ Yes. You might want to get hold of some textbook on Poisson processes... $\endgroup$ – Did Oct 13 '14 at 10:50
  • $\begingroup$ Ok, thanks. Any recommendation? $\endgroup$ – Alexander Oct 13 '14 at 10:55
  • $\begingroup$ Sure, type poisson processes lecture notes in your favorite search engine, and rejoice. $\endgroup$ – Did Oct 13 '14 at 13:37
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@Did actually answered this same question: Conditioning a Poisson process on the number of arrivals in a fixed time

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 $$\mathbb E[E_3] = \sum_{i=1}^3\mathbb E[E_i-E_{i-1}]= 3.$$

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