CHI SQUARE DISTRIBUTION

Phitter implementation

Distribution Definition

import phitter

distribution = phitter.continuous.ChiSquare({"df": *})

💡 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

$$X\sim {\chi }^2\left(\text{df}\right)$$

Distribution Domain

$$x\in (0,\mathrm{\infty })$$

Parameters Domain and Constraints

$$\text{df}\in {\mathbb{N}}^{+}$$

Cumulative Distribution Function

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

Probability Density Function

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

Percent Point Function / Sample

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

Parametric Centered Moments

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

Parametric Mean

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

Parametric Variance

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

Parametric Skewness

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

Parametric Kurtosis

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

Parametric Median

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

Parametric Mode

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

Additional Information and Definitions

• $u:\text{Uniform[0,1] random varible}$
• $\text{P}(a,x)=\frac{\gamma (a,x)}{\mathrm{\Gamma }(a)}:\text{Regularized lower incomplete gamma function}$
• ${\text{P}}^{-1}(a,u):\text{Inverse of regularized lower incomplete gamma function}$
• $\gamma (a,x):\text{Lower incomplete gamma function}$
• $\mathrm{\Gamma }\left(x\right):\text{Gamma function}$

Spreadsheet Documents

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