CAUCHY DISTRIBUTION

Phitter implementation

Distribution Definition

import phitter

distribution = phitter.continuous.Cauchy({"x0": *, "gamma": *})

💡 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

$$X\sim \mathrm{Cauchy}(x0,\gamma )$$

Distribution Domain

$$x\in (-\mathrm{\infty },+\mathrm{\infty })$$

Parameters Domain and Constraints

$$x_0\in \mathbb{R},\gamma \in {\mathbb{R}}^{+}$$

Cumulative Distribution Function

$$F_X\left(x\right)=\frac{1}{\pi }\arctan \left(\frac{x-x_0}{\gamma }\right)+\frac{1}{2}$$

Probability Density Function

$$f_X\left(x\right)=\frac{1}{\pi \gamma [1+{\left(\frac{x-x_0}{\gamma }\right)}^2]}$$

Percent Point Function / Sample

$$F_X^{-1}\left(u\right)=x_0+\gamma \tan \left[\pi (u-\frac{1}{2})\right]$$

Parametric Centered Moments

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

Parametric Mean

$$\mathrm{Mean}(X)=\text{undefined}$$

Parametric Variance

$$\mathrm{Variance}(X)=\text{undefined}$$

Parametric Skewness

$$\mathrm{Skewness}(X)=\text{undefined}$$

Parametric Kurtosis

$$\mathrm{Kurtosis}(X)=\text{undefined}$$

Parametric Median

$$\mathrm{Median}(X)=x_0$$

Parametric Mode

$$\mathrm{Mode}(X)=x_0$$

Additional Information and Definitions

• $x_0:\text{Location parameter}$
• $\gamma :\text{Scale parameter}$
• $u:\text{Uniform[0,1] random varible}$

Spreadsheet Documents

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