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. 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

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

for some C > 0. 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

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

for all t > 0.

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

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

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