F DISTRIBUTION

Phitter implementation

Distribution Definition

import phitter

distribution = phitter.continuous.F({"df1": *, "df2": *})

💡 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

$$X\sim \mathrm{F}({\text{df}}_1,{\text{df}}_2)$$

Distribution Domain

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

Parameters Domain and Constraints

$${\text{df}}_1\in {\mathbb{R}}^{+},{\text{df}}_2\in {\mathbb{R}}^{+}$$

Cumulative Distribution Function

$$F_X\left(x\right)=I_{{\text{df}}_{1}x/({\text{df}}_{1}x+{\text{df}}_2)}(\frac{{\text{df}}_1}{2},\frac{{\text{df}}_2}{2})$$

Probability Density Function

$$f_X\left(x\right)=\frac{\sqrt{\frac{({\text{df}}_{1}x{)}^{{\text{df}}_1}{\text{df}}_2^{{\text{df}}_2}}{({\text{df}}_{1}x+{\text{df}}_2{)}^{{\text{df}}_1+{\text{df}}_2}}}}{x\times \text{Beta}(\frac{{\text{df}}_1}{2},\frac{{\text{df}}_2}{2})}$$

Percent Point Function / Sample

$$F_X^{-1}\left(u\right)=\frac{{\text{df}}_2\times I^{-1}(u,\frac{{\text{df}}_1}{2},\frac{{\text{df}}_2}{2})}{{\text{df}}_1\times (1-I^{-1}(u,\frac{{\text{df}}_1}{2},\frac{{\text{df}}_2}{2}))}$$

Parametric Centered Moments

$${\mu }_k^{\prime }=E[X^k]={\int }_0^{\mathrm{\infty }}{x}^k{f}_X\left(x\right)dx={\left(\frac{{\text{df}}_2}{{\text{df}}_1}\right)}^k\frac{\mathrm{\Gamma }(\frac{{\text{df}}_1}{2}+k)}{\mathrm{\Gamma }\left(\frac{{\text{df}}_1}{2}\right)}\frac{\mathrm{\Gamma }(\frac{{\text{df}}_2}{2}-k)}{\mathrm{\Gamma }\left(\frac{{\text{df}}_2}{2}\right)}\text{if }{\text{df}}_2>2k$$

Parametric Mean

$$\mathrm{Mean}(X)={\mu }_1^{\prime }=\frac{{\text{df}}_2}{{\text{df}}_2-2}\text{if }{\text{df}}_2>2$$

Parametric Variance

$$\mathrm{Variance}(X)={\mu }_2^{\prime }-{\mu }_1^{\prime 2}=\frac{2{\text{df}}_2^2({\text{df}}_1+{\text{df}}_2-2)}{{\text{df}}_1({\text{df}}_2-2{)}^2({\text{df}}_2-4)}\text{if }{\text{df}}_2>4$$

Parametric Skewness

$$\mathrm{Skewness}(X)=\frac{{\mu }_3^{\prime }-3{\mu }_2^{\prime }{\mu }_1^{\prime }+2{\mu }_1^{\prime 3}}{({\mu }_2^{\prime }-{\mu }_1^{\prime 2}{)}^{1.5}}=\frac{(2{\text{df}}_1+{\text{df}}_2-2)\sqrt{8({\text{df}}_2-4)}}{({\text{df}}_2-6)\sqrt{{\text{df}}_1({\text{df}}_1+{\text{df}}_2-2)}}\text{if }{\text{df}}_2>6$$

Parametric Kurtosis

$$\mathrm{Kurtosis}(X)=\frac{{\mu }_4^{\prime }-4{\mu }_1^{\prime }{\mu }_3^{\prime }+6{\mu }_1^{\prime 2}{\mu }_2^{\prime }-3{\mu }_1^{\prime 4}}{({\mu }_2^{\prime }-{\mu }_1^{\prime 2}{)}^2}=\frac{3(8+({\text{df}}_2-6)\times \mathrm{Skewness}(X{)}^2)}{2{\text{df}}_2-16}+3\text{if }{\text{df}}_2>8$$

Parametric Median

$$\mathrm{Median}(X)=\frac{{\text{df}}_2\times I^{-1}(\frac{1}{2},\frac{{\text{df}}_1}{2},\frac{{\text{df}}_2}{2})}{{\text{df}}_1\times (1-I^{-1}(\frac{1}{2},\frac{{\text{df}}_1}{2},\frac{{\text{df}}_2}{2}))}$$

Parametric Mode

$$\mathrm{Mode}(X)=\frac{{\text{df}}_2({\text{df}}_1-2)}{{\text{df}}_1({\text{df}}_2+2)}\text{if }{\text{df}}_1>2$$

Additional Information and Definitions

• $u:\text{Uniform[0,1] random varible}$
• $I(x,a,b):\text{Regularized incomplete beta function}$
• $I^{-1}(x,a,b):\text{Inverse of regularized incomplete beta function}$
• $\text{Beta}(x,y):\text{Beta function}$

Spreadsheet Documents

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