Sub-Poissonian

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_ exp(t X)\le E_ exp(C t X)

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_ exp(t X)\le E_ exp(t X)= \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 \le \exp(p(e^t-1)).
Because sub-Poissonianism is preserved by sums, we get that the binomial distribution is also sub-Poissonian.

References

Poisson point processes
Types of probability distributions

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