R-distribution Distribution

A general-purpose distribution with a variety of shapes controlled by \(c>0.\) Range of standard distribution is \(x\in\left[-1,1\right]\)

\[ \begin{eqnarray*} f\left(x;c\right) & = & \frac{\left(1-x^{2}\right)^{c/2-1}}{B\left(\frac{1}{2},\frac{c}{2}\right)}\\ F\left(x;c\right) & = & \frac{1}{2}+\frac{x}{B\left(\frac{1}{2},\frac{c}{2}\right)}\,_{2}F_{1}\left(\frac{1}{2},1-\frac{c}{2};\frac{3}{2};x^{2}\right)\end{eqnarray*}\]

\[\mu_{n}^{\prime}=\frac{\left(1+\left(-1\right)^{n}\right)}{2}B\left(\frac{n+1}{2},\frac{c}{2}\right)\]

The R-distribution with parameter \(n\) is the distribution of the correlation coefficient of a random sample of size \(n\) drawn from a bivariate normal distribution with \(\rho=0.\) The mean of the standard distribution is always zero and as the sample size grows, the distribution’s mass concentrates more closely about this mean.

Implementation: scipy.stats.rdist