# Anyone recognize this single parameter discrete probability distribution?

1. Jun 14, 2010

### jacobcdf

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 $$E(k^{2}) = L$$.

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

Does anyone recognize this distribution?

J.

Last edited: Jun 14, 2010
2. Jun 14, 2010

### jacobcdf

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 solution, as the asymptotic approximation appears of little use practically.

J.

3. Jun 15, 2010

### jacobcdf

In case anyone's interested, this distribution appears to be a special case of the Conway–Maxwell–Poisson distribution with the non-Poisson parameter $$\nu = 2$$.