The Skellam distribution is the discrete probability distribution of the difference N 1 − N 2 {\displaystyle N_{1}-N_{2}} of two statistically independent random variables N 1 {\displaystyle N_{1}} and N 2 , {\displaystyle N_{2},} each Poisson-distributed with respective expected values μ 1 {\displaystyle \mu _{1}} and μ 2 {\displaystyle \mu _{2}} . It is useful in describing the statistics of the difference of two images with simple photon noise, as well as describing the point spread distribution in sports where all scored points are equal, such as baseball, hockey and soccer. The distribution is also applicable to a special case of the difference of dependent Poisson random variables, but just the obvious case where the two variables have a common additive random contribution which is cancelled by the differencing: see Karlis & Ntzoufras (2003) for details and an application. The probability mass function for the Skellam distribution for a difference K = N 1 − N 2 {\displaystyle K=N_{1}-N_{2}} between two independent Poisson-distributed random variables with means μ 1 {\displaystyle \mu _{1}} and μ 2 {\displaystyle \mu _{2}} is given by:
p ( k ; μ 1 , μ 2 ) = Pr { K = k } = e − ( μ 1 + μ 2 ) ( μ 1 μ 2 ) k / 2 I k ( 2 μ 1 μ 2 ) {\displaystyle p(k;\mu _{1},\mu _{2})=\Pr\{K=k\}=e^{-(\mu _{1}+\mu _{2})}\left({\mu _{1} \over \mu _{2}}\right)^{k/2}I_{k}(2{\sqrt {\mu _{1}\mu _{2}}})}
where Ik(z) is the modified Bessel function of the first kind. Since k is an integer we have that Ik(z) = I|k|(z).
Derivation The probability mass function of a Poisson-distributed random variable with mean μ is given by
p ( k ; μ ) = μ k k ! e − μ . {\displaystyle p(k;\mu )={\mu ^{k} \over k!}e^{-\mu }.\,}
for k ≥ 0 {\displaystyle k\geq 0} (and zero otherwise). The Skellam probability mass function for the difference of two independent counts K = N 1 − N 2 {\displaystyle K=N_{1}-N_{2}} is the convolution of two Poisson distributions: (Skellam, 1946)
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