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Anyone recognize this single parameter discrete probability distribution?

  1. Jun 14, 2010 #1
    I have a single parameter discrete probability distribution defined over the domain of non-negative integers with pmf in k of:

    [tex]Pr(k;L) = \frac{L^{k}}{k! * k! * I_{0}(2*\sqrt{L})}[/tex]

    Where [tex]I_{0}()[/tex] is the modified Bessel function of the first kind with order 0.

    I do know that [tex]E(k^{2}) = L[/tex].

    Can anyone come up with a closed form for the distribution mean?

    Does anyone recognize this distribution?

    Thanks in advance,
    Last edited: Jun 14, 2010
  2. jcsd
  3. Jun 14, 2010 #2
    I also know that as [tex]L \rightarrow \infty[/tex]:

    [tex]\gamma_{1} \rightarrow \sqrt{\frac{1}{2*\sqrt{K}}[/tex]


    [tex]\gamma_{2} \rightarrow \gamma_{1}^{2}[/tex]

    and E(k) appears to approach something approximated by:

    [tex]\sqrt{L - \frac{\sqrt{L}}{2}}[/tex]

    But regardless, I still would like an exact closed-form solution, as the asymptotic approximation appears of little use practically.

  4. Jun 15, 2010 #3
    In case anyone's interested, this distribution appears to be a special case of the Conway–Maxwell–Poisson distribution with the non-Poisson parameter [tex]\nu = 2[/tex].
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