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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?

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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 $n(1-e^{-2a})$.

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  • $\begingroup$ There is no "approximately" needed if instead you consider the cumulative sums of an infinite series of iid exponential variables: they are a Poisson process on the positive real numbers. $\endgroup$
    – whuber
    Feb 20, 2021 at 15:27

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