Super-Poissonian distribution

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

${\displaystyle E_{X\sim D}[\exp(tX)]\leq E_{X\sim E}[\exp(CtX)].}$

for some C > 0.[3] This implies that if ${\displaystyle X_{1}}$ and ${\displaystyle X_{2}}$ are both from a sub-E distribution, then so is ${\displaystyle X_{1}+X_{2}}$.

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

${\displaystyle E_{X\sim D}[\exp(tX)]\leq E_{X\sim {\text{Poisson}}(\lambda )}[\exp(tX)]=\exp(\lambda (e^{t}-1)),}$

for all t > 0.[4]

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

${\displaystyle E[\exp(tX)]=(1-p)+pe^{t}\leq \exp(p(e^{t}-1)).}$

Because sub-Poissonianism is preserved by sums, we get that the binomial distribution is also sub-Poissonian.

References

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