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
formula_1
for some "C ; 0".
This implies that if formula_2 and formula_3 are both from a sub-E distribution, then so is formula_4.
A distribution is "strictly sub-" if "C ≤ 1".
From this definition a distribution, "D", is sub-Poissonian if
formula_5
for all "t ; 0".
An example of a sub-Poissonian distribution is the Bernoulli distribution, since
formula_6
Because sub-Poissonianism is preserved by sums, we get that the binomial distribution is also sub-Poissonian.
References.
 ;templatestyles src="Reflist/styles.css" / ;