Question

How to show inter arrival times are exponential in poisson distribution?

Answer 1

The binomial distribution deals with the number of successes in a fixed number of independent trials, and the geometric distribution deals with the time between successes in a series of independent trials. Just so, the Poisson distribution deals with the number of occurrences in a fixed period of time, and the exponential distribution deals with the time between occurrences of successive events as time flows by continuously
If the mean interarrival time is 1/ (so  is the mean arrival rate per unit time), then the variance will be 1/2 (and the standard deviation will be 1/ ). The graph below displays the graph of the exponential density function when  = 1 . Generally, if X is exponentially distributed, then Pr(s < X  t) = e-s - e-t (where e  2.71828) .