T STUDENT 3P DISTRIBUTION

Phitter implementation

Distribution Definition

import phitter

distribution = phitter.continuous.TStudent3P({"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

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

Distribution Domain

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

Parameters Domain and Constraints

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

Cumulative Distribution Function

$$F_X\left(x\right)=I(\frac{z(x)+\sqrt{z(x{)}^2+\text{df}}}{2\sqrt{z(x{)}^2+\text{df}}},\frac{\text{df}}{2},\frac{\text{df}}{2})$$

Probability Density Function

$$f_X\left(x\right)=\frac{{(1+z(x{)}^2/\text{df})}^{-(1+\text{df})/2}}{\sqrt{\text{df}}\times \text{Beta}(\frac{1}{2},\frac{\text{df}}{2})}$$

Percent Point Function / Sample

$$F_X^{-1}\left(u\right)=\{\begin{array}{cl}\text{Loc}+\text{Sc}\sqrt{\frac{\text{df}(1-I^{-1}(u,\text{df}/2,\text{df}/2))}{I^{-1}(u,\text{df}/2,\text{df}/2)}}& \text{if }u\ge \frac{1}{2}\\ \text{Loc}-\text{Sc}\sqrt{\frac{\text{df}(1-I^{-1}(u,\text{df}/2,\text{df}/2))}{I^{-1}(u,\text{df}/2,\text{df}/2)}}& \text{if }u<\frac{1}{2}\end{array}$$

Parametric Centered Moments

$${\tilde{\mu }}_k^{\prime }=E[{\tilde{X}}^k]={\int }_0^{\mathrm{\infty }}{x}^k{f}_{\tilde{X}}\left(x\right)dx=\{\begin{array}{cl}0& \text{if }k\text{ odd}\wedge 0<k<\text{df}\\ {\text{df}}^{\frac{k}{2}}\prod _{i=1}^{k/2}\frac{2i-1}{\text{df}-2i}& \text{if }k\text{ even}\wedge 0<k<\text{df}\end{array}$$

Parametric Mean

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

Parametric Variance

$$\mathrm{Variance}(X)={\text{Sc}}^2\times ({\tilde{\mu }}_2^{\prime }-{\tilde{\mu }}_1^{\prime 2})=\{\begin{array}{cl}{\text{Sc}}^2\text{df}/(\text{df}+2)& \text{if }\text{df}>2\\ \text{undefined}& \text{if }\text{df}\le 2\end{array}$$

Parametric Skewness

$$\mathrm{Skewness}(X)=\frac{{\tilde{\mu }}_3^{\prime }-3{\tilde{\mu }}_2^{\prime }{\tilde{\mu }}_1^{\prime }+2{\tilde{\mu }}_1^{\prime 3}}{({\tilde{\mu }}_2^{\prime }-{\tilde{\mu }}_1^{\prime 2}{)}^{1.5}}=\{\begin{array}{cl}0& \text{if }\text{df}>3\\ \text{undefined}& \text{if }\text{df}\le 3\end{array}$$

Parametric Kurtosis

$$\mathrm{Kurtosis}(X)=\frac{{\tilde{\mu }}_4^{\prime }-4{\tilde{\mu }}_1^{\prime }{\tilde{\mu }}_3^{\prime }+6{\tilde{\mu }}_1^{\prime 2}{\tilde{\mu }}_2^{\prime }-3{\tilde{\mu }}_1^{\prime 4}}{({\tilde{\mu }}_2^{\prime }-{\tilde{\mu }}_1^{\prime 2}{)}^2}=\{\begin{array}{cl}3+6/(\text{df}-4)& \text{if }\text{df}>4\\ \text{undefined}& \text{if }\text{df}\le 4\end{array}$$

Parametric Median

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

Parametric Mode

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

Additional Information and Definitions

• $\tilde{X}\sim \mathrm{TStudent}\left(\text{df}\right)$
• $\text{Loc}:\text{Location parameter}$
• $\text{Sc}:\text{Scale parameter}$
• $z\left(x\right)=(x-\text{Loc})/\text{Sc}$
• $u:\text{Uniform[0,1] random varible}$
• $I(x,a,b):\text{Regularized incomplete beta function}$
• $I^{-1}(x,a,b):\text{Inverse of regularized incomplete beta function}$
• $\text{Beta}(x,y):\text{Beta function}$

Spreadsheet Documents

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