In case anyone's interested, this distribution appears to be a special case of the Conway–Maxwell–Poisson distribution with the non-Poisson parameter \nu = 2.
I also know that as L \rightarrow \infty:
\gamma_{1} \rightarrow \sqrt{\frac{1}{2*\sqrt{K}}
and
\gamma_{2} \rightarrow \gamma_{1}^{2}
and E(k) appears to approach something approximated by:
\sqrt{L - \frac{\sqrt{L}}{2}}
But regardless, I still would like an exact closed-form...
I have a single parameter discrete probability distribution defined over the domain of non-negative integers with pmf in k of:
Pr(k;L) = \frac{L^{k}}{k! * k! * I_{0}(2*\sqrt{L})}
Where I_{0}() is the modified Bessel function of the first kind with order 0.
I do know that E(k^{2}) = L...
Muchos gracias, amigo. That helps a lot. :)
Although now that I think about I should have been able to intuit that one on my own. I'm just getting old I guess. :(
Hi all,
I've been messing around with the product of Poisson distributions and was hoping someone could help me work out a closed form solution for the following convergent infinite series (given x > 0):
\sum^{\infty}_{i=0} \frac{x^{i}}{i! \times i!}
Many thanks in advance,
Jacob.