CHI SQUARE 3P DISTRIBUTION

Phitter implementation

Distribution Definition

import phitter

distribution = phitter.continuous.ChiSquare3P({"df": *, "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

$$X\sim {\chi }_{3\mathrm{P}}^2(\text{df},\text{Loc},\text{Sc})$$

Distribution Domain

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

Parameters Domain and Constraints

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

Cumulative Distribution Function

$$F_X\left(x\right)=\frac{\gamma (\frac{\text{df}}{2},\frac{z(x)}{2})}{\mathrm{\Gamma }(\frac{\text{df}}{2})}=\text{P}(\frac{\text{df}}{2},\frac{z(x)}{2})$$

Probability Density Function

$$f_X\left(x\right)=\frac{1}{\text{Sc}}\frac{1}{2^{\text{df}/2}\mathrm{\Gamma }(\text{df}/2)}{x}^{\text{df}/2-1}{e}^{-z(x)/2}$$

Percent Point Function / Sample

$$F_X^{-1}\left(u\right)=2{\text{P}}^{-1}(\frac{\text{df}}{2},u)$$

Parametric Centered Moments

$${\tilde{\mu }}_k^{\prime }=E[{\tilde{X}}^k]={\int }_0^{\mathrm{\infty }}{x}^k{f}_{\tilde{X}}\left(x\right)dx=\text{df}(\text{df}+2)\cdots (\text{df}+2k-2)=2^k\frac{\mathrm{\Gamma }(k+\frac{\text{df}}{2})}{\mathrm{\Gamma }\left(\frac{\text{df}}{2}\right)}$$

Parametric Mean

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

Parametric Variance

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

Parametric Skewness

$$\mathrm{Skewness}(X)=\frac{{\tilde{\mu }}_3^{\prime }-3{\tilde{\mu }}_2^{\prime }{\tilde{\mu }}_1^{\prime }+2{\tilde{\mu }}_1^{\prime 3}}{({\tilde{\mu }}_2^{\prime }-{\tilde{\mu }}_1^{\prime 2}{)}^{1.5}}=\sqrt{\frac{8}{\text{df}}}$$

Parametric Kurtosis

$$\mathrm{Kurtosis}(X)=\frac{{\tilde{\mu }}_4^{\prime }-4{\tilde{\mu }}_1^{\prime }{\tilde{\mu }}_3^{\prime }+6{\tilde{\mu }}_1^{\prime 2}{\tilde{\mu }}_2^{\prime }-3{\tilde{\mu }}_1^{\prime 4}}{({\tilde{\mu }}_2^{\prime }-{\tilde{\mu }}_1^{\prime 2}{)}^2}=3+\frac{12}{\text{df}}$$

Parametric Median

$$\mathrm{Median}(X)=\text{Loc}+\text{Sc}\times 2{\text{P}}^{-1}(\frac{\text{df}}{2},\frac{1}{2})$$

Parametric Mode

$$\mathrm{Mode}(X)=\text{Loc}+\text{Sc}\times \text{max}(\text{df}-2,0)$$

Additional Information and Definitions

• $\tilde{X}\sim {\chi }^2\left(\text{df}\right)$
• $\text{Loc}:\text{Location parameter}$
• $\text{Sc}:\text{Scale parameter}$
• $z\left(x\right)=(x-\text{Loc})/\text{Sc}$
• $u:\text{Uniform[0,1] random varible}$
• $\text{P}(a,x)=\frac{\gamma (a,x)}{\mathrm{\Gamma }(a)}:\text{Regularized lower incomplete gamma function}$
• ${\text{P}}^{-1}(a,u):\text{Inverse of regularized lower incomplete gamma function}$
• $\gamma (a,x):\text{Lower incomplete gamma function}$
• $\mathrm{\Gamma }\left(x\right):\text{Gamma function}$

Spreadsheet Documents

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