Editing Skellam distribution (section)

== Derivation ==

The [[probability mass function]] of a [[Poisson distribution|Poisson-distributed]] random variable with mean μ is given by

:<math>
 p(k;\mu)={\mu^k\over k!}e^{-\mu}.\,
 </math>

for <math>k \ge 0</math> (and zero otherwise). The Skellam probability mass function for the difference of two independent counts <math>K=N_1-N_2</math> is the [[convolution]] of two Poisson distributions: ([[John Gordon Skellam|Skellam]], 1946)

:<math>
\begin{align}
  p(k;\mu_1,\mu_2)
  & =\sum_{n=-\infty}^\infty
  p(k+n;\mu_1)p(n;\mu_2) \\
  & =e^{-(\mu_1+\mu_2)}\sum_{n=\max(0,-k)}^\infty
  {{\mu_1^{k+n}\mu_2^n}\over{n!(k+n)!}}
\end{align}
  </math>
Since the Poisson distribution is zero for negative values of the count <math>(p(N<0;\mu)=0)</math>, the second sum is only taken for those terms where <math>n\ge0</math> and <math>n+k\ge0</math>. It can be shown that the above sum implies that

:<math>\frac{p(k;\mu_1,\mu_2)}{p(-k;\mu_1,\mu_2)}=\left(\frac{\mu_1}{\mu_2}\right)^k</math>

so that:

:<math>
  p(k;\mu_1,\mu_2)= e^{-(\mu_1+\mu_2)}
  \left({\mu_1\over\mu_2}\right)^{k/2}I_{|k|}(2\sqrt{\mu_1\mu_2})
  </math>

where ''I''&nbsp;<sub>k</sub>(z) is the [[Bessel function#Modified Bessel functions : I.CE.B1.2C K.CE.B1|modified Bessel function]] of the first kind. The special case for <math>\mu_1=\mu_2(=\mu)</math> is given by Irwin (1937):

:<math>
  p\left(k;\mu,\mu\right) = e^{-2\mu}I_{|k|}(2\mu).
 </math>

Using the limiting values of the modified Bessel function for small arguments, we can recover the Poisson distribution as a special case of the Skellam distribution for <math>\mu_2=0</math>.