SEMICIRCULAR DISTRIBUTION

Phitter implementation

Distribution Definition

import phitter

distribution = phitter.continuous.Semicircular({"loc": *, "R": *})

💡 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

$$X\sim \mathrm{Semicircular}(\text{Loc},R)$$

Distribution Domain

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

Parameters Domain and Constraints

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

Cumulative Distribution Function

$$F_X\left(x\right)=\frac{1}{2}+\frac{z(x)\sqrt{R^2-z(x{)}^2}}{\pi R^2}+\frac{\arcsin \left(\frac{z(x)}{R}\right)}{\pi }$$

Probability Density Function

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

Percent Point Function / Sample

$$F_X^{-1}\left(u\right)=\text{Loc}+R\times (2I^{-1}(u,1.5,1.5)-1)$$

Parametric Centered Moments

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

Parametric Mean

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

Parametric Variance

$$\mathrm{Variance}(X)={\mu }_2^{\prime }-{\mu }_1^{\prime 2}=\frac{R^2}{4}$$

Parametric Skewness

$$\mathrm{Skewness}(X)=\frac{{\mu }_3^{\prime }-3{\mu }_2^{\prime }{\mu }_1^{\prime }+2{\mu }_1^{\prime 3}}{({\mu }_2^{\prime }-{\mu }_1^{\prime 2}{)}^{1.5}}=0$$

Parametric Kurtosis

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

Parametric Median

$$\mathrm{Median}(X)=\text{Loc}$$

Parametric Mode

$$\mathrm{Mode}(X)=\text{Loc}$$

Additional Information and Definitions

• $\text{Loc}:\text{Location parameter}$
• $R:\text{Scale parameter}$
• $z\left(x\right)=x-\text{Loc}$
• $u:\text{Uniform[0,1] random varible}$
• $I^{-1}(x,a,b):\text{Inverse of regularized incomplete beta function}$

Spreadsheet Documents

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