This is documentation for an old release of SciPy (version 1.3.0). Read this page in the documentation of the latest stable release (version 1.15.1).
Reciprocal Distribution¶
There are two shape parameters \(a,b>0\) and the support is \(x\in\left[a,b\right]\).
\begin{eqnarray*} f\left(x;a,b\right) & = & \frac{1}{x\log\left(b/a\right)}\\ F\left(x;a,b\right) & = & \frac{\log\left(x/a\right)}{\log\left(b/a\right)}\\ G\left(q;a,b\right) & = & a\exp\left(q\log\left(b/a\right)\right)=a\left(\frac{b}{a}\right)^{q}\end{eqnarray*}
\begin{eqnarray*} d & = & \log\left(a/b\right)\\ \mu & = & \frac{a-b}{d}\\ \mu_{2} & = & \mu\frac{a+b}{2}-\mu^{2}=\frac{\left(a-b\right)\left[a\left(d-2\right)+b\left(d+2\right)\right]}{2d^{2}}\\ \gamma_{1} & = & \frac{\sqrt{2}\left[12d\left(a-b\right)^{2}+d^{2}\left(a^{2}\left(2d-9\right)+2abd+b^{2}\left(2d+9\right)\right)\right]}{3d\sqrt{a-b}\left[a\left(d-2\right)+b\left(d+2\right)\right]^{3/2}}\\ \gamma_{2} & = & \frac{-36\left(a-b\right)^{3}+36d\left(a-b\right)^{2}\left(a+b\right)-16d^{2}\left(a^{3}-b^{3}\right)+3d^{3}\left(a^{2}+b^{2}\right)\left(a+b\right)}{3\left(a-b\right)\left[a\left(d-2\right)+b\left(d+2\right)\right]^{2}}-3\\ m_{d} & = & a\\ m_{n} & = & \sqrt{ab}\end{eqnarray*}
\[h\left[X\right]=\frac{1}{2}\log\left(ab\right)+\log\left[\log\left(\frac{b}{a}\right)\right].\]
Implementation: scipy.stats.reciprocal