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Re: Skellam distribution

Here's the PDF from Mathematica,

f1 = PDF[d1 = PoissonDistribution[m1]];
f2 = PDF[d2 = PoissonDistribution[m2]];
pdf[k_] = Sum[f1[k + n] f2[n], {n, -k, Infinity}] // PowerExpand

E^(-m1 - m2) m1^(k/2) m2^(-k/2) BesselI[-k, 2 Sqrt[m1] Sqrt[m2]]

  verified at

and here's the CDF:

messy[k_] = Sum[pdf[n],{n,0,k}]

E^(-m1-m2) (-((Sqrt[m2] BesselI[1,2 Sqrt[m1]  
\[FormalY][1+\[FormalN]]+(1+\[FormalN]-m2) \[FormalY][2+\[FormalN]]+m2  
BesselI[1,2 Sqrt[m1] Sqrt[m2]])/Sqrt[m1],\[FormalY][1]==BesselI[0,2  
Sqrt[m1] Sqrt[m2]]+(Sqrt[m2] BesselI[1,2 Sqrt[m1]  

That doesn't look nice at all, so I'd probably use

cdf[-1] = 0;
cdf[k_] := cdf[k] = cdf[k - 1] + pdf[k]

messy[10] - cdf[10] // FullSimplify



On Fri, 31 Jul 2009 04:52:36 -0500, Peter Breitfeld <phbrf at>  

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

DrMajorBob at

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