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