Fit inter-arrival time to Poisson Distribution/Exponential

Question

I am trying to fit the Poisson distribution to my data which is the inter-arrival times. The format of the data is:

time, count

2,3

5,1

8,2

,which means I had 3 objects with the inter-arrival time of 2 seconds, 1 object with 5 sec inter-arrival time, and so on. I a bit confused that I shall fit the Exponential distribution or Poisson distribution to these data as Poisson models the counts and Exponentials models the time between inter-arrival times.

I am using R and I am using fitdistrplus library. So, I change the data as follows 2 2 2 5 8 8 and then fit these data with fitdistrplus.

It seems that my approach agrees with : How do you fit a Poisson distribution to table data?

But just want to be sure that I need to fit my data with Exponential distribution not with Poisson.

Answer 1

The Poisson distribution is for event counts in a given interval of time (in a Poisson process). It is not useful for modelling inter-event times.

If the events follow a Poisson process, the inter-event times will be distributed as exponential.

The usual (ML, MoM) estimator of the exponential mean is the sample mean; the ML estimator of the rate (the reciprocal of the population mean) is the reciprocal of the sample mean.