INVERSE GAMMA DISTRIBUTION

Phitter implementation

Distribution Definition

import phitter

distribution = phitter.continuous.InverseGamma({"alpha": *, "beta": *})

💡 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

$$X\sim \mathrm{InverseGamma}(\alpha ,\beta )$$

Distribution Domain

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

Parameters Domain and Constraints

$$\alpha \in {\mathbb{R}}^{+},\beta \in {\mathbb{R}}^{+}$$

Cumulative Distribution Function

$$F_X\left(x\right)=1-\frac{\gamma (\alpha ,\beta /x)}{\mathrm{\Gamma }(\alpha )}=1-\text{P}(\alpha ,\frac{\beta }{x})$$

Probability Density Function

$$f_X\left(x\right)=\frac{{\beta }^{\alpha }}{\mathrm{\Gamma }(\alpha )}{x}^{-\alpha -1}\exp (-\frac{\beta }{x})$$

Percent Point Function / Sample

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

Parametric Centered Moments

$${\tilde{\mu }}_k^{\prime }=E[{\tilde{X}}^k]={\int }_0^{\mathrm{\infty }}{x}^k{f}_{\tilde{X}}\left(x\right)dx=\frac{\mathrm{\Gamma }(\alpha -k)}{\mathrm{\Gamma }(\alpha )}=\frac{1}{(\alpha -1)\cdots (\alpha -k)}\text{if }\alpha >k$$

Parametric Mean

$$\mathrm{Mean}(X)=\beta {\tilde{\mu }}_1^{\prime }$$

Parametric Variance

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

Parametric Skewness

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

Parametric Kurtosis

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

Parametric Median

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

Parametric Mode

$$\mathrm{Mode}(X)=\frac{\beta }{\alpha +1}$$

Additional Information and Definitions

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

Spreadsheet Documents

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