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Folded Normal Distribution

If $Z$ is Normal with mean $L$ and $\sigma=S$ , then $\left|Z\right|$ is a folded normal with shape parameter $c=\left|L\right|/S$ , location parameter $0$ and scale parameter $S$ . This is a special case of the non-central chi distribution with one- degree of freedom and non-centrality parameter $c^{2}.$ Note that $c\geq0$ . The standard form of the folded normal is

$$\begin{eqnarray*} f\left(x;c\right) & = & \sqrt{\frac{2}{\pi}}\cosh\left(cx\right)\exp\left(-\frac{x^{2}+c^{2}}{2}\right)\\ F\left(x;c\right) & = & \Phi\left(x-c\right)-\Phi\left(-x-c\right)=\Phi\left(x-c\right)+\Phi\left(x+c\right)-1\\ G\left(\alpha;c\right) & = & F^{-1}\left(x;c\right)\end{eqnarray*}$$

$$M\left(t\right)=\exp\left[\frac{t}{2}\left(t-2c\right)\right]\left(1+e^{2ct}\right)$$

$$\begin{eqnarray*} k & = & \mathrm{erf}\left(\frac{c}{\sqrt{2}}\right)\\ p & = & \exp\left(-\frac{c^{2}}{2}\right)\\ \mu & = & \sqrt{\frac{2}{\pi}}p+ck\\ \mu_{2} & = & c^{2}+1-\mu^{2}\\ \gamma_{1} & = & \frac{\sqrt{\frac{2}{\pi}}p^{3}\left(4-\frac{\pi}{p^{2}}\left(2c^{2}+1\right)\right)+2ck\left(6p^{2}+3cpk\sqrt{2\pi}+\pi c\left(k^{2}-1\right)\right)}{\pi\mu_{2}^{3/2}}\\ \gamma_{2} & = & \frac{c^{4}+6c^{2}+3+6\left(c^{2}+1\right)\mu^{2}-3\mu^{4}-4p\mu\left(\sqrt{\frac{2}{\pi}}\left(c^{2}+2\right)+\frac{ck}{p}\left(c^{2}+3\right)\right)}{\mu_{2}^{2}}\end{eqnarray*}$$

Implementation: scipy.stats.foldnorm