Re: Skellam distribution

• To: mathgroup at smc.vnet.net
• Subject: [mg102208] Re: [mg102152] Skellam distribution
• From: Bob Hanlon <hanlonr at cox.net>
• Date: Sat, 1 Aug 2009 04:01:11 -0400 (EDT)

```Clear[SkellamDistribution];

SkellamDistribution::usage =
"SkellamDistribution[a1,a2] is the Skellam discrete probability \
distribution with parameters a1 and a2. \
http://en.wikipedia.org/wiki/Skellam_distribution";;

SkellamDistribution /: PDF[
SkellamDistribution[a1_, a2_], k_] :=

Exp[-(a1 + a2)]*(a1/a2)^(k/2)*
BesselI[k, 2 Sqrt[a1*a2]];

SkellamDistribution /: mgf[
SkellamDistribution[a1_, a2_], t_] :=

Exp[-(a1 + a2) + a1*Exp[t] + a2*Exp[-t]];

SkellamDistribution /: CharacteristicFunction[
SkellamDistribution[a1_, a2_], t_] :=

Exp[-(a1 + a2) + a1*Exp[I*t] + a2*Exp[-I*t]];

SkellamDistribution /: Moment[
SkellamDistribution[a1_, a2_], n_Integer] :=
Module[{t},
D[mgf[SkellamDistribution[a1, a2], t], {t, n}] /.
t -> 0];

SkellamDistribution /: CDF[
SkellamDistribution[a1_, a2_], k_] :=
Sum[PDF[
SkellamDistribution[a1, a2], x],
{x, -Infinity, k}];

SkellamDistribution /: Mean[
SkellamDistribution[a1_, a2_]] :=
(a1 - a2);

SkellamDistribution /: Variance[
SkellamDistribution[a1_, a2_]] :=
(a1 + a2);

SkellamDistribution /: StandardDeviation[
SkellamDistribution[a1_, a2_]] :=
Sqrt[a1 + a2];

Table[{n,
Moment[SkellamDistribution[a1, a2], n]},
{n, 0, 2}]

{{0, 1}, {1, a1 - a2}, {2, (a1 - a2)^2 + a1 + a2}}

Bob Hanlon

---- Peter Breitfeld <phbrf at t-online.de> wrote:

=============

Is there an implementation of the Skellam distribution? If not, how
would you implement a CDF for this distribution?

--
_________________________________________________________________
Peter Breitfeld, Bad Saulgau, Germany -- http://www.pBreitfeld.de

```

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