Super-Poissonian distribution explained

In mathematics, a super-Poissonian distribution is a probability distribution that has a larger variance than a Poisson distribution with the same mean.^[1] Conversely, a sub-Poissonian distribution has a smaller variance.

An example of super-Poissonian distribution is negative binomial distribution.^[2]

The Poisson distribution is a result of a process where the time (or an equivalent measure) between events has an exponential distribution, representing a memoryless process.

Mathematical definition

In probability theory it is common to say a distribution, D, is a sub-distribution of another distribution E if D 's moment-generating function, is bounded by E 's up to a constant.In other words

E _{X\sim} [\exp(t X)] \le E _{X\sim} [\exp(C t X)]. for some C > 0.^[3] This implies that if X _{1} and X _{2} are both from a sub-E distribution, then so is X _{1+X} _{2} .

A distribution is strictly sub- if C ≤ 1.From this definition a distribution, D, is sub-Poissonian if

E _{X\sim} [\exp(t X)] \le E _{X\sim} [\exp(t X)] = \exp( λ (e ^{t-1)),} for all t > 0.^[4]

An example of a sub-Poissonian distribution is the Bernoulli distribution, since

E[\exp(t X)] = ( 1- p) + p e ^{t} \le \exp(p(e ^{t-1)).} Because sub-Poissonianism is preserved by sums, we get that the binomial distribution is also sub-Poissonian.

Notes and References

1. Zou . X. . Mandel . L. . 10.1103/PhysRevA.41.475 . Photon-antibunching and sub-Poissonian photon statistics . Physical Review A . 41 . 1 . 475–476 . 1990 . 9902890. 1990PhRvA..41..475Z .
2. Anders . Simon . Huber . Wolfgang . 10.1186/gb-2010-11-10-r106 . Differential expression analysis for sequence count data . Genome Biology . 11 . R106 . 2010 . 10 . 3218662 . 20979621 . free .
3. Book: Vershynin, Roman . High-Dimensional Probability: An Introduction with Applications in Data Science . 2018-09-27 . Cambridge University Press . 978-1-108-24454-1 . en.
4. Ahle . Thomas D. . 2022-03-01 . Sharp and simple bounds for the raw moments of the binomial and Poisson distributions . Statistics & Probability Letters . en . 182 . 109306 . 10.1016/j.spl.2021.109306 . 2103.17027 . 0167-7152.