NAKAGAMI DISTRIBUTION

Phitter implementation

Distribution Definition

import phitter

distribution = phitter.continuous.Nakagami({"m": *, "omega": *})

💡 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

$$X\sim \mathrm{Nakagami}(m,\mathrm{\Omega })$$

Distribution Domain

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

Parameters Domain and Constraints

$$m\in {\mathbb{R}}_{⩾\frac{1}{2}}^{+},\mathrm{\Omega }\in {\mathbb{R}}^{+}$$

Cumulative Distribution Function

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

Probability Density Function

$$f_X\left(x\right)=\frac{2m^m}{\mathrm{\Gamma }(m){\mathrm{\Omega }}^m}{x}^{2m-1}\exp (-\frac{m}{\mathrm{\Omega }}{x}^2)$$

Percent Point Function / Sample

$$F_X^{-1}\left(u\right)=\sqrt{\frac{\mathrm{\Omega }}{m}{\text{P}}^{-1}(m,u)}$$

Parametric Centered Moments

$${\mu }_k^{\prime }=E[X^k]={\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}{x}^k{f}_X\left(x\right)dx$$

Parametric Mean

$$\mathrm{Mean}(X)={\mu }_1^{\prime }=\frac{\mathrm{\Gamma }(m+\frac{1}{2})}{\mathrm{\Gamma }(m)}{\left(\frac{\mathrm{\Omega }}{m}\right)}^{1/2}$$

Parametric Variance

$$\mathrm{Variance}(X)={\mu }_2^{\prime }-{\mu }_1^{\prime 2}=\mathrm{\Omega }(1-\frac{1}{m}{\left(\frac{\mathrm{\Gamma }(m+\frac{1}{2})}{\mathrm{\Gamma }(m)}\right)}^2)$$

Parametric Skewness

$$\mathrm{Skewness}(X)=\frac{{\mu }_3^{\prime }-3{\mu }_2^{\prime }{\mu }_1^{\prime }+2{\mu }_1^{\prime 3}}{({\mu }_2^{\prime }-{\mu }_1^{\prime 2}{)}^{1.5}}=\frac{\frac{\mathrm{\Gamma }(m+\frac{1}{2})}{\mathrm{\Gamma }(m)\sqrt{m}}(1-4m(1-\frac{1}{m}{\left(\frac{\mathrm{\Gamma }(m+\frac{1}{2})}{\mathrm{\Gamma }(m)}\right)}^2))}{2m{(1-\frac{1}{m}{\left(\frac{\mathrm{\Gamma }(m+\frac{1}{2})}{\mathrm{\Gamma }(m)}\right)}^2)}^{3/2}}$$

Parametric Kurtosis

$$\mathrm{Kurtosis}(X)=\frac{{\mu }_4^{\prime }-4{\mu }_1^{\prime }{\mu }_3^{\prime }+6{\mu }_1^{\prime 2}{\mu }_2^{\prime }-3{\mu }_1^{\prime 4}}{({\mu }_2^{\prime }-{\mu }_1^{\prime 2}{)}^2}=3+\frac{-6{\left(\frac{\mathrm{\Gamma }(m+\frac{1}{2})}{\mathrm{\Gamma }(m)\sqrt{m}}\right)}^{4}m+(8m-2){\left(\frac{\mathrm{\Gamma }(m+\frac{1}{2})}{\mathrm{\Gamma }(m)\sqrt{m}}\right)}^2-2m+1}{m{(1-\frac{1}{m}{\left(\frac{\mathrm{\Gamma }(m+\frac{1}{2})}{\mathrm{\Gamma }(m)}\right)}^2)}^2}$$

Parametric Median

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

Parametric Mode

$$\mathrm{Mode}(X)=\frac{\sqrt{2}}{2}{\left(\frac{(2m-1)\mathrm{\Omega }}{m}\right)}^{1/2}$$

Additional Information and Definitions

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

Spreadsheet Documents

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