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

[math]\displaystyle{ E_{X\sim D}[\exp(t X)] \le E_{X\sim E}[\exp(C t X)]. }[/math]

for some C > 0. This implies that if [math]\displaystyle{ X_1 }[/math] and [math]\displaystyle{ X_2 }[/math] are both from a sub-E distribution, then so is [math]\displaystyle{ X_1+X_2 }[/math].

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

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

for all t > 0.

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

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

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