Proof of Exponential Interarrival times times

[mmehdi]

The proof that poisson process has exponential interarrival time is common place. The proof which i am trying to do is that, exponential interarrival times will always be poisson process, its like the reverse of the earlier proof. Could you help me with that.

 

[HallsofIvy]

The fundamental assumption of the Poisson distribution is that, for some very small time interval, the probability of a single arrival is a constant and the probability of more than one arrival in that time interval is so small it can be ignored. The proof that that hypothesis leads to the Poisson distribution is given in any good probability text that discusses the Poison distribution. You can also treat that as a differential equations problem (the rate of change of total arrivals is constant) that has an exponential function as solution.

 

[mmehdi]

I am not sure if I truly follow you. The proof that you can only have one arrival in one interval and the probability of getting two arrival is zero, is done through taylor series expansion. But how does it show that the exponenttial interarrival shall always satisfy the poisson properties.

What I tried is the total time, or waiting time let's say for the second arrival is Gamma Distribution. So if we integrate the gamma distribution it shall give us a poisson distribution. However the integration is winding, but it does yield the poisson distribution at the end.