Continuous Poisson Distribution

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Is there a Continuous analogous of the Poisson Distribution? Under the analogous, I mean such a distribution that:

1. It is a one-parameter distribution
2. Its distribution function is similar to the Poisson one

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1 Answer

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There are different types of continuous extensions of Poisson distribution.

I favor the one presented in this paper:

"Continuous counterparts of Poisson and binomial distributions and their properties" by Andrii Ilienko.

They are based on integral representations of Poisson and binomial probability distribution functions using complete and incomplete (Euler) $\Gamma$ and $\operatorname{B}$ functions.

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