FOLDED NORMAL DISTRIBUTION

Phitter implementation

Distribution Definition

import phitter

distribution = phitter.continuous.FoldedNormal({"mu": *, "sigma": *})

💡 The distribution's parameters are defined equation section below

Distribution Methods and Attributes

## CDF, PDF, PPF receive float or numpy.ndarray.
distribution.cdf(float | numpy.ndarray) # -> float | numpy.ndarray
distribution.pdf(float | numpy.ndarray) # -> float | numpy.ndarray
distribution.ppf(float | numpy.ndarray) # -> float | numpy.ndarray
distribution.sample(int) # -> numpy.ndarray

## STATS
distribution.mean # -> float
distribution.variance # -> float
distribution.standard_deviation # -> float
distribution.skewness # -> float
distribution.kurtosis # -> float
distribution.median # -> float
distribution.mode # -> float

Equations

Distribution Definition

$$X\sim \mathrm{FoldedNormal}(\mu ,\sigma )$$

Distribution Domain

$$x\in [0,\mathrm{\infty })$$

Parameters Domain and Constraints

$$\mu \in \mathbb{R},\sigma \in {\mathbb{R}}^{+}$$

Cumulative Distribution Function

$$F_X\left(x\right)=\frac{1}{2}[\text{erf}\left(\frac{x+\mu }{\sigma \sqrt{2}}\right)+\text{erf}\left(\frac{x-\mu }{\sigma \sqrt{2}}\right)]$$

Probability Density Function

$$f_X\left(x\right)=\frac{1}{\sigma \sqrt{2\pi }}{e}^{-\frac{(x-\mu {)}^2}{2{\sigma }^2}}+\frac{1}{\sigma \sqrt{2\pi }}{e}^{-\frac{(x+\mu {)}^2}{2{\sigma }^2}}$$

Percent Point Function / Sample

$${\text{Sample}}_X\left(u\right)=|\mu +\sigma {\mathrm{\Phi }}^{-1}(u)|$$

Parametric Centered Moments

$${\mu }_k^{\prime }=E[X^k]={\int }_0^{\mathrm{\infty }}{x}^k{f}_X\left(x\right)dx$$

Parametric Mean

$$\mathrm{Mean}(X)={\mu }_1^{\prime }=\sigma \sqrt{\frac{2}{\pi }}{e}^{(-{\mu }^2/2{\sigma }^2)}+\mu (1-2\mathrm{\Phi }(-\frac{\mu }{\sigma }))$$

Parametric Variance

$$\mathrm{Variance}(X)={\mu }_2^{\prime }-{\mu }_1^{\prime 2}={\mu }^2+{\sigma }^2-\mathrm{Mean}(X{)}^2$$

Parametric Skewness

$$\mathrm{Skewness}(X)=\frac{{\mu }_3^{\prime }-3{\mu }_2^{\prime }{\mu }_1^{\prime }+2{\mu }_1^{\prime 3}}{({\mu }_2^{\prime }-{\mu }_1^{\prime 2}{)}^{1.5}}$$

Parametric Kurtosis

$$\mathrm{Kurtosis}(X)=\frac{{\mu }_4^{\prime }-4{\mu }_1^{\prime }{\mu }_3^{\prime }+6{\mu }_1^{\prime 2}{\mu }_2^{\prime }-3{\mu }_1^{\prime 4}}{({\mu }_2^{\prime }-{\mu }_1^{\prime 2}{)}^2}$$

Parametric Median

$$\mathrm{Median}(X)=|\mu +\sigma {\mathrm{\Phi }}^{-1}\left(1/2\right)|$$

Parametric Mode

$$\mathrm{Mode}(X)=\arg \underset{x}{max}{f}_X\left(x\right)$$

Additional Information and Definitions

• $$\begin{gathered} \text{Computing an analytic expression for the inverse of the cumulative distribution function is not} \\ \text{feasible. Nonetheless, it is possible to generate a random sample from the distribution.} \end{gathered}$$
• $\mu :\text{Location parameter}$
• $\sigma :\text{Scale parameter}$
• $u:\text{Uniform[0,1] random varible}$
• $\mathrm{\Phi }\left(x\right):\text{CDF normal standard distribution}$
• $\varphi \left(x\right):\text{PDF normal standard distribution}$
• ${\mathrm{\Phi }}^{-1}\left(x\right):\text{PPF normal standard distribution}$

Spreadsheet Documents

• Phitter playground
• Download Excel Spreadsheet
• Excel file from GitHub repository
• Google spreadsheet document