HALF NORMAL DISTRIBUTION

Phitter implementation

Distribution Definition

import phitter

distribution = phitter.continuous.HalfNormal({"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

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

Distribution Domain

$$x\in (\mu ,\mathrm{\infty })$$

Parameters Domain and Constraints

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

Cumulative Distribution Function

$$F_X\left(x\right)=2\mathrm{\Phi }(z(x))-1=\mathrm{erf}\left(\frac{z(x)}{\sqrt{2}}\right)$$

Probability Density Function

$$f_X\left(x\right)=\frac{\sqrt{2}}{\sigma \sqrt{\pi }}\exp (-\frac{z(x{)}^2}{2})$$

Percent Point Function / Sample

$$F_X^{-1}\left(u\right)=\mu +\sigma {\mathrm{\Phi }}^{-1}\left(\frac{1+u}{2}\right)=\tilde{\mu }+\sigma \sqrt{2}{\mathrm{erf}}^{-1}(u)$$

Parametric Centered Moments

$${\tilde{\mu }}_k^{\prime }=E[{\tilde{X}}^k]={\int }_0^{\mathrm{\infty }}{x}^k{f}_{\tilde{X}}\left(x\right)dx=\frac{2^{n/2}\mathrm{\Gamma }(\frac{n+1}{2})}{\sqrt{\pi }}$$

Parametric Mean

$$\mathrm{Mean}(X)=\tilde{\mu }+\sigma {\tilde{\mu }}_1^{\prime }=\tilde{\mu }+\sigma \sqrt{\frac{2}{\pi }}$$

Parametric Variance

$$\mathrm{Variance}(X)={\sigma }^2({\tilde{\mu }}_2^{\prime }-{\tilde{\mu }}_1^{\prime 2})={\sigma }^2(1-\frac{2}{\pi })$$

Parametric Skewness

$$\mathrm{Skewness}(X)=\frac{{\tilde{\mu }}_3^{\prime }-3{\tilde{\mu }}_2^{\prime }{\tilde{\mu }}_1^{\prime }+2{\tilde{\mu }}_1^{\prime 3}}{({\tilde{\mu }}_2^{\prime }-{\tilde{\mu }}_1^{\prime 2}{)}^{1.5}}=\frac{\sqrt{2}(4-\pi )}{(\pi -2{)}^{3/2}}=0.9952717$$

Parametric Kurtosis

$$\mathrm{Kurtosis}(X)=\frac{{\tilde{\mu }}_4^{\prime }-4{\tilde{\mu }}_1^{\prime }{\tilde{\mu }}_3^{\prime }+6{\tilde{\mu }}_1^{\prime 2}{\tilde{\mu }}_2^{\prime }-3{\tilde{\mu }}_1^{\prime 4}}{({\tilde{\mu }}_2^{\prime }-{\tilde{\mu }}_1^{\prime 2}{)}^2}=3+\frac{8(\pi -3)}{(\pi -2{)}^2}=3.869177$$

Parametric Median

$$\mathrm{Median}(X)=\mu +\sigma \sqrt{2}{\mathrm{erf}}^{-1}(1/2)$$

Parametric Mode

$$\mathrm{Mode}(X)=\mu$$

Additional Information and Definitions

• $\tilde{X}\sim \mathrm{HalfNormal}(0,1)$
• $\mu :\text{Location parameter}$
• $\sigma :\text{Scale parameter}$
• $z\left(x\right)=(x-\mu )/\sigma$
• $u:\text{Uniform[0,1] random varible}$
• $\mathrm{\Phi }\left(x\right):\text{CDF normal standard distribution}$
• ${\mathrm{\Phi }}^{-1}\left(x\right):\text{PPF normal standard distribution}$
• $\mathrm{erf}(x):\text{Error function}$
• $\mathrm{\Gamma }\left(x\right):\text{Gamma function}$

Spreadsheet Documents

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