Editing Skellam distribution (section)

== Properties ==

As it is a discrete probability function, the Skellam probability mass function is normalized:

:<math>
  \sum_{k=-\infty}^\infty p(k;\mu_1,\mu_2)=1.
  </math>

We know that the [[Probability-generating function|probability generating function]] (pgf) for a [[Poisson distribution]] is:

:<math>
  G\left(t;\mu\right)= e^{\mu(t-1)}.
  </math>

It follows that the pgf, <math>G(t;\mu_1,\mu_2)</math>, for a Skellam probability mass function will be:

:<math>
\begin{align}
G(t;\mu_1,\mu_2) & = \sum_{k=-\infty}^\infty p(k;\mu_1,\mu_2)t^k \\[4pt]
& = G\left(t;\mu_1\right)G\left(1/t;\mu_2\right) \\[4pt]
& = e^{-(\mu_1+\mu_2)+\mu_1 t+\mu_2/t}.
\end{align}
</math>

Notice that the form of the [[probability-generating function]] implies that the distribution of the sums or the differences of any number of independent Skellam-distributed variables are again Skellam-distributed. It is sometimes claimed that any linear combination of two Skellam distributed variables are again Skellam-distributed, but this is clearly not true since any multiplier other than <math>\pm 1</math> would change the [[support (mathematics)|support]] of the distribution and alter the pattern of [[Moment (mathematics)|moments]] in a way that no Skellam distribution can satisfy.

The [[moment-generating function]] is given by:

:<math>M\left(t;\mu_1,\mu_2\right) = G(e^t;\mu_1,\mu_2) = \sum_{k=0}^\infty { t^k \over k!}\,m_k</math>

which yields the raw moments ''m''<sub>''k''</sub>&nbsp;. Define:

:<math>\Delta\ \stackrel{\mathrm{def}}{=}\  \mu_1-\mu_2\,</math>

:<math>\mu\ \stackrel{\mathrm{def}}{=}\   (\mu_1+\mu_2)/2.\,</math>

Then the raw moments ''m''<sub>''k''</sub> are

:<math>m_1=\left.\Delta\right.\,</math>

:<math>m_2=\left.2\mu+\Delta^2\right.\,</math>

:<math>m_3=\left.\Delta(1+6\mu+\Delta^2)\right.\,</math>

The [[moment about the mean|central moments]] ''M''<sub> ''k''</sub> are

:<math>M_2=\left.2\mu\right.,\,</math>

:<math>M_3=\left.\Delta\right.,\,</math>

:<math>M_4=\left.2\mu+12\mu^2\right..\,</math>

The [[expected value|mean]], [[variance]], [[skewness]], and [[kurtosis|kurtosis excess]] are respectively:

:<math>
\begin{align}
\operatorname E(n) & = \Delta, \\[4pt]
\sigma^2 & =2\mu, \\[4pt]
\gamma_1 & =\Delta/(2\mu)^{3/2}, \\[4pt]
\gamma_2 & = 1/2.
\end{align}
</math>

The [[cumulant-generating function]] is given by:

:<math>
  K(t;\mu_1,\mu_2)\ \stackrel{\mathrm{def}}{=}\   \ln(M(t;\mu_1,\mu_2))
  = \sum_{k=0}^\infty { t^k \over k!}\,\kappa_k
  </math>

which yields the [[cumulant]]s:

:<math>\kappa_{2k}=\left.2\mu\right.</math>
:<math>\kappa_{2k+1}=\left.\Delta\right. .</math>

For the special case when μ<sub>1</sub> = μ<sub>2</sub>, an
[[asymptotic expansion]] of the [[Bessel function|modified Bessel function of the first kind]] yields for large μ:

:<math>
  p(k;\mu,\mu)\sim
  {1\over\sqrt{4\pi\mu}}\left[1+\sum_{n=1}^\infty
  (-1)^n{\{4k^2-1^2\}\{4k^2-3^2\}\cdots\{4k^2-(2n-1)^2\}
  \over n!\,2^{3n}\,(2\mu)^n}\right].
  </math>

(Abramowitz & Stegun 1972, p.&nbsp;377).  Also, for this special case, when ''k'' is also large, and of [[Big O notation|order]] of the square root of 2μ, the distribution tends to a [[normal distribution]]:

:<math>
  p(k;\mu,\mu)\sim
  {e^{-k^2/4\mu}\over\sqrt{4\pi\mu}}.
  </math>

These special results can easily be extended to the more general case of different means.

===Bounds on weight above zero===
If <math>X \sim \operatorname{Skellam} (\mu_1, \mu_2) </math>, with <math>\mu_1 < \mu_2</math>, then

::<math>
\frac{\exp(-(\sqrt{\mu_1} -\sqrt{\mu_2})^2  )}{(\mu_1 + \mu_2)^2} - \frac{e^{-(\mu_1 + \mu_2)}}{2\sqrt{\mu_1 \mu_2}} - \frac{e^{-(\mu_1 + \mu_2)}}{4\mu_1 \mu_2} \leq \Pr\{X  \geq 0\} \leq \exp (- (\sqrt{\mu_1} -\sqrt{\mu_2})^2)
</math>

Details can be found in [[Poisson distribution#Poisson races]]