Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

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,
    J.
     
    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]

    and

    [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.


    J.
     
  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].
     
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook




Loading...