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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 $\lambda$.

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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

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