Hermite distribution

Hermite
Probability mass function The horizontal axis is the index k, the number of occurrences. The function is only defined at integer values of k. The connecting lines are only guides for the eye.
Cumulative distribution function The horizontal axis is the index k, the number of occurrences. The CDF is discontinuous at the integers of k and flat everywhere else because a variable that is Hermite distributed only takes on integer values.
Notation	${\displaystyle \operatorname {Herm} (a_{1},a_{2})\,}$
Parameters	a1 ≥ 0, a2 ≥ 0
Support	x ∈ { 0, 1, 2, ... }
PMF	${\displaystyle x\mapsto e^{-(a_{1}+a_{2})}\sum _{j=0}^{\lfloor x/2\rfloor }{\frac {a_{1}^{x-2j}a_{2}^{j}}{(x-2j)!j!}}}$
CDF	${\displaystyle x\mapsto e^{-a_{1}+a_{2}}\sum _{i=0}^{\lfloor x\rfloor }\sum _{j=0}^{\lfloor i/2\rfloor }{\frac {a_{1}^{i-2j}a_{2}^{j}}{(i-2j)!j!}}}$
Mean	${\displaystyle a_{1}+2a_{2}}$
Variance	${\displaystyle a_{1}+4a_{2}}$
Skewness	${\displaystyle {\frac {a_{1}+8a_{2}}{(a_{1}+4a_{2})^{3/2}}}}$
Excess kurtosis	${\displaystyle {\frac {a_{1}+16a_{2}}{(a_{1}+4a_{2})^{2}}}}$
MGF	${\displaystyle \exp(a_{1}(e^{t}-1)+a_{2}(e^{2t}-1))\,}$
CF	${\displaystyle \exp(a_{1}(e^{ti}-1)+a_{2}(e^{2ti}-1))\,}$
PGF	${\displaystyle \exp(a_{1}(s-1)+a_{2}(s^{2}-1))\,}$

In probability theory and statistics, the Hermite distribution, named after Charles Hermite, is a discrete probability distribution used to model count data with more than one parameter. This distribution is flexible in terms of its ability to allow a moderate over-dispersion in the data.

The authors C. D. Kemp and A. W. Kemp have called it "Hermite distribution" from the fact its probability function and the moment generating function can be expressed in terms of the coefficients of (modified) Hermite polynomials.