# Poisson variate corresponding to the Exponential variate

According to Wikipedia, In probability theory and statistics, the exponential distribution is the probability distribution of the time between events in a Poisson point process, i.e., a process in which events occur continuously and independently at a constant average rate.

A certain example from the book on Statistical Inference by George and Casella is as follows, Let X1,.....Xn be a random sample from an exponential (a) population. Specifically, X1,....Xn might correspond to the times until failure (measured in years) for n identical circuit boards that are put on test and used until they fail. P(X>2) is the probability that the board will last more than 2 years.

What is the corresponding poisson variate for the exponential variate, X?

If $$n$$ is large and you have a bunch of identical things that could fail indeepdently from each other and the probability that each one of them fails is small, then the number of failures within a fixed period of time is approximately Poisson. Take the 2 year time period for example and rate $$a$$. The probability that a single circuit board will last more than one year is $$e^{-2a}$$. The probability that it will fail within the first 2 years is $$1-e^{-2a}$$. The number (out of all $$n$$ circuit boards) that will fail within the first 2 years is Binomial with mean $$n(1-e^{-2a})$$. If $$n$$ is large, then this will be approximately Poisson with mean $$n(1-e^{-2a})$$.