GIBRAT DISTRIBUTION

Phitter implementation

Distribution Definition

import phitter

distribution = phitter.continuous.Gibrat({"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

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

Distribution Domain

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

Parameters Domain and Constraints

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

Cumulative Distribution Function

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

Probability Density Function

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

Percent Point Function / Sample

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

Parametric Centered Moments

$${\mu }_k^{\prime }=E[X^k]={\int }_{\text{Loc}}^{\mathrm{\infty }}{x}^k{f}_X\left(x\right)dx=\exp \left(\frac{k^2}{2}\right)$$

Parametric Mean

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

Parametric Variance

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

Parametric Skewness

$$\mathrm{Skewness}(X)=\frac{{\mu }_3^{\prime }-3{\mu }_2^{\prime }{\mu }_1^{\prime }+2{\mu }_1^{\prime 3}}{({\mu }_2^{\prime }-{\mu }_1^{\prime 2}{)}^{1.5}}=\sqrt{e-1}(2+e)$$

Parametric Kurtosis

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

Parametric Median

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

Parametric Mode

$$\mathrm{Mode}(X)=\text{Loc}+\frac{\text{Sc}}{e}$$

Additional Information and Definitions

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

Spreadsheet Documents

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