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Super-Poissonian distribution

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

An example of super-Poissonian distribution is negative binomial distribution.

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 ∼ D [ exp ⁡ ( t X ) ] ≤ E X ∼ E [ exp ⁡ ( C t X ) ] . {\displaystyle E_{X\sim D}\leq E_{X\sim E}.}

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

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

E X ∼ D [ exp ⁡ ( t X ) ] ≤ E X ∼ Poisson ( λ ) [ exp ⁡ ( t X ) ] = exp ⁡ ( λ ( e t − 1 ) ) , {\displaystyle E_{X\sim D}\leq E_{X\sim {\text{Poisson}}(\lambda )}=\exp(\lambda (e^{t}-1)),}

for all t > 0.

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

E [ exp ⁡ ( t X ) ] = ( 1 − p ) + p e t ≤ exp ⁡ ( p ( e t − 1 ) ) . {\displaystyle E=(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.


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