# Poisson from Exponential

The probability distribution of time between events following a Poisson point process with parameter $$\lambda$$ is an Exponential Distribution with parameter $$\lambda$$.

The proof from Poisson to Exponential is a standard one which can be found in basic statistics references.

However, is it possible to derive the Poisson from the Exponential, i.e.

Given a random process with time between events which follow an Exponential Distribution with parameter $$\lambda$$, show that the distribution of occurrences of events is a Poisson Distribution with parameter $$\lambda$$.

Hint: In the proof of deriving Exponential from Poisson, you can observe that the Poisson distribution with $$n$$ occurrence is equal to the distance between two tail probability of Erlang distribution with parameters $$(n,\lambda)$$ and $$(n+1,\lambda)$$ Some further algebraic manipulation should give you the proof