Skellam distribution information

Skellam
	Probability mass function; Examples of the probability mass function for the Skellam distribution. The horizontal axis is the index k. (The function is only defined at integer values of k. The connecting lines do not indicate continuity.)
Parameters
Support
PMF
Mean
Median	N/A
Variance
Skewness
Excess kurtosis
MGF
CF

The Skellam distribution is the discrete probability distribution of the difference $N_{1}-N_{2}$ of two statistically independent random variables $N_{1}$ and $N_{2},$ each Poisson-distributed with respective expected values $\mu _{1}$ and $\mu _{2}$ . 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 $K=N_{1}-N_{2}$ between two independent Poisson-distributed random variables with means $\mu _{1}$ and $\mu _{2}$ is given by:

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 I_k(z) is the modified Bessel function of the first kind. Since k is an integer we have that I_k(z)=I_|k|(z).

Skellam distribution information

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Probability mass function Examples of the probability mass function for the Skellam distribution. The horizontal axis is the index k. (The function is only defined at integer values of k. The connecting lines do not indicate continuity.)
Parameters	$\mu _{1}\geq 0,~~\mu _{2}\geq 0$
Support	$k\in \{\ldots ,-2,-1,0,1,2,\ldots \}$
PMF	$e^{-(\mu _{1}\!+\!\mu _{2})}\left({\frac {\mu _{1}}{\mu _{2}}}\right)^{k/2}\!\!I_{k}(2{\sqrt {\mu _{1}\mu _{2}}})$
Mean	$\mu _{1}-\mu _{2}\,$
Median	N/A
Variance	$\mu _{1}+\mu _{2}\,$
Skewness	${\frac {\mu _{1}-\mu _{2}}{(\mu _{1}+\mu _{2})^{3/2}}}$
Excess kurtosis	${\frac {1}{\mu _{1}+\mu _{2}}}$
MGF	$e^{-(\mu _{1}+\mu _{2})+\mu _{1}e^{t}+\mu _{2}e^{-t}}$
CF	$e^{-(\mu _{1}+\mu _{2})+\mu _{1}e^{it}+\mu _{2}e^{-it}}$