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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