ALPHA DISTRIBUTION

Phitter implementation

Distribution Definition

import phitter

distribution = phitter.continuous.Alpha({"alpha": *, "loc": *, "scale": *})

💡 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

$$X\sim \mathrm{Alpha}(\alpha ,\text{Loc},\text{Sc})$$

Distribution Domain

$$x\in (\text{Loc},\mathrm{\infty })$$

Parameters Domain and Constraints

$$\alpha \in {\mathbb{R}}^{+},\text{Loc}\in \mathbb{R},\text{Sc}\in {\mathbb{R}}^{+}$$

Cumulative Distribution Function

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

Probability Density Function

$$f_X\left(x\right)=\frac{1}{\text{Sc}\cdot z(x{)}^2\cdot \mathrm{\Phi }\left(\alpha \right)\cdot \sqrt{2\pi }}\exp (-\frac{1}{2}{(\alpha -\frac{1}{z(x)})}^2)$$

Percent Point Function / Sample

$$F_X^{-1}\left(u\right)=\text{Loc}+\text{Sc}\times \frac{1}{\alpha -{\mathrm{\Phi }}^{-1}\left(u\mathrm{\Phi }\left(\alpha \right)\right)}$$

Parametric Centered Moments

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

Parametric Mean

$$\mathrm{Mean}(X)=\text{Loc}+\text{Sc}\cdot {\tilde{\mu }}_1^{\prime }$$

Parametric Variance

$$\mathrm{Variance}(X)={\text{Sc}}^2\cdot ({\tilde{\mu }}_2^{\prime }-{\tilde{\mu }}_1^{\prime 2})$$

Parametric Skewness

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

Parametric Kurtosis

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

Parametric Median

$$\mathrm{Median}(X)=\text{Loc}+\frac{\text{Sc}}{\alpha -{\mathrm{\Phi }}^{-1}\left(\frac{1}{2}\mathrm{\Phi }\left(\alpha \right)\right)}$$

Parametric Mode

$$\mathrm{Mode}(X)=\text{Loc}+\text{Sc}\frac{(\sqrt{{\alpha }^2+8}-\alpha )}{4}$$

Additional Information and Definitions

• $\tilde{X}\sim \mathrm{Alpha}(\alpha ,0,1)$
• $\text{Loc}:\text{Location parameter}$
• $\text{Sc}:\text{Scale parameter}$
• $z(x)=(x-\text{Loc})/\text{Sc}$
• $u:\text{Uniform[0,1] random varible}$
• $\mathrm{\Phi }\left(x\right):\text{CDF normal standard distribution}$
• ${\mathrm{\Phi }}^{-1}\left(x\right):\text{PPF normal standard distribution}$

Spreadsheet Documents

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