BETA PRIME 4P DISTRIBUTION

Phitter implementation

Distribution Definition

import phitter

distribution = phitter.continuous.BetaPrime4P({"alpha": *, "beta": *, "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

$$X\sim {\mathrm{BetaPrime}}_{4\mathrm{P}}(\alpha ,\beta ,\text{Loc},\text{Sc})$$

Distribution Domain

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

Parameters Domain and Constraints

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

Cumulative Distribution Function

$$F_X\left(x\right)=I(\frac{z(x)}{1+z(x)},\alpha ,\beta )$$

Probability Density Function

$$f_X\left(x\right)=\frac{z(x{)}^{\alpha -1}(1+z(x){)}^{-\alpha -\beta }}{\text{Sc}\times \text{Beta}(\alpha ,\beta )}$$

Percent Point Function / Sample

$$F_X^{-1}\left(u\right)=\text{Loc}+\text{Sc}\frac{I^{-1}(u,\alpha ,\beta )}{1-I^{-1}(u,\alpha ,\beta )}$$

Parametric Centered Moments

$${\tilde{\mu }}_k^{\prime }=E[{\tilde{X}}^k]={\int }_0^{\mathrm{\infty }}{x}^k{f}_{\tilde{X}}\left(x\right)dx=\frac{\mathrm{\Gamma }(k+\alpha )\mathrm{\Gamma }(\beta -k)}{\mathrm{\Gamma }\left(\alpha \right)\mathrm{\Gamma }\left(\beta \right)}\text{if }\beta >k$$

Parametric Mean

$$\mathrm{Mean}(X)=\text{Loc}+\text{Sc}{\tilde{\mu }}_1^{\prime }=\text{Loc}+\text{Sc}\frac{\alpha }{\beta -1}\text{if }\beta >1$$

Parametric Variance

$$\mathrm{Variance}(X)={\text{Sc}}^2({\tilde{\mu }}_2^{\prime }-{\tilde{\mu }}_1^{\prime 2})={\text{Sc}}^2\frac{\alpha (\alpha +\beta -1)}{(\beta -2)(\beta -1{)}^2}\text{if }\beta >2$$

Parametric Skewness

$$\mathrm{Skewness}(X)=\frac{{\tilde{\mu }}_3^{\prime }-3{\tilde{\mu }}_2^{\prime }{\tilde{\mu }}_1^{\prime }+2{\tilde{\mu }}_1^{\prime 3}}{({\tilde{\mu }}_2^{\prime }-{\tilde{\mu }}_1^{\prime 2}{)}^{1.5}}=\frac{2(2\alpha +\beta -1)}{\beta -3}\sqrt{\frac{\beta -2}{\alpha (\alpha +\beta -1)}}\text{if }\beta >3$$

Parametric Kurtosis

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

Parametric Median

$$\mathrm{Median}(X)=\text{Loc}+\text{Sc}\frac{I^{-1}(\frac{1}{2},\alpha ,\beta )}{1-I^{-1}(\frac{1}{2},\alpha ,\beta )}$$

Parametric Mode

$$\mathrm{Mode}(X)=\text{Loc}+\text{Sc}\frac{\alpha -1}{\beta +1}$$

Additional Information and Definitions

• $\tilde{X}\sim \mathrm{BetaPrime}(\alpha ,\beta )$
• $\text{Loc}:\text{Location parameter}$
• $\text{Sc}:\text{Scale parameter}$
• $z\left(x\right)=(x-\text{Loc})/\text{Sc}$
• $u:\text{Uniform[0,1] random varible}$
• $I(x,a,b):\text{Regularized incomplete beta function}$
• $I^{-1}(x,a,b):\text{Inverse of regularized incomplete beta function}$
• $\mathrm{\Gamma }\left(x\right):\text{Gamma function}$
• $\text{Beta}(x,y):\text{Beta function}$

Spreadsheet Documents

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