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 a super-Poissonian distribution is the 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 ∼ D [ exp ( t X ) ] ≤ E X ∼ E [ exp ( C t X ) ] . {\displaystyle E_{X\sim D}[\exp(tX)]\leq E_{X\sim E}[\exp(CtX)].}
for some C > 0. This implies that if X 1 {\displaystyle X_{1}} and X 2 {\displaystyle X_{2}} are both from a sub-E distribution, then so is X 1 + X 2 {\displaystyle X_{1}+X_{2}} . A distribution is strictly sub- if C ≤ 1. From this definition a distribution, D, is sub-Poissonian if
E X ∼ D [ exp ( t X ) ] ≤ E X ∼ Poisson ( λ ) [ exp ( t X ) ] = exp ( λ ( e t − 1 ) ) , {\displaystyle 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 ( t X ) ] = ( 1 − p ) + p e t ≤ exp ( p ( e t − 1 ) ) . {\displaystyle 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.
References