In mathematics, asuper-Poissonian distribution is aprobability distribution that has a largervariance than aPoisson distribution with the samemean.^[1] Conversely, asub-Poissonian distribution has a smaller variance.

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

ThePoisson distribution is a result of a process where the time (or an equivalent measure) between events has anexponential distribution, representing amemoryless process.

Inprobability theory it is common to say a distribution,D, is a sub-distribution of another distributionE ifD 'smoment-generating function, is bounded byE 's up to a constant.In other words

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

for someC > 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 isstrictly sub- ifC ≤ 1.From this definition a distribution,D, is sub-Poissonian if

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

for allt > 0.^[4]

An example of a sub-Poissonian distribution is theBernoulli distribution, since

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

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

Thisphysics-related article is astub. You can help Wikipedia byexpanding it.

• v
• t
• e

• Poisson point processes
• Types of probability distributions
• Physics stubs

• All stub articles