Rice Distribution

There is one shape parameter \(b\geq0\) (the “distance from the origin”) and the support is \(x\geq0\).

\begin{eqnarray*} f\left(x;b\right) & = & x\exp\left(-\frac{x^{2}+b^{2}}{2}\right)I_{0}\left(xb\right)\\ F\left(x;b\right) & = & \int_{0}^{x}\alpha\exp\left(-\frac{\alpha^{2}+b^{2}}{2}\right)I_{0}\left(\alpha b\right)d\alpha\end{eqnarray*}

were \(I_{0}(y)\) is the modified Bessel function of the first kind of order 0.

\[\mu_{n}^{\prime}=\sqrt{2^{n}}\Gamma\left(1+\frac{n}{2}\right)\,_{1}F_{1}\left(-\frac{n}{2};1;-\frac{b^{2}}{2}\right)\]

Implementation: scipy.stats.rice