RICE DISTRIBUTION

Phitter implementation

Distribution Definition

import phitter

distribution = phitter.continuous.Rice({"v": *, "sigma": *})

💡 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

$$X\sim \mathrm{Rice}(v,\sigma )$$

Distribution Domain

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

Parameters Domain and Constraints

$$v\in {\mathbb{R}}^{+},\sigma \in {\mathbb{R}}^{+}$$

Cumulative Distribution Function

$$F_X\left(x\right)=1-Q_1(\frac{v}{\sigma },\frac{x}{\sigma })$$

Probability Density Function

$$f_X\left(x\right)=\frac{x}{{\sigma }^2}\exp \left(\frac{-(x^2+v^2)}{2{\sigma }^2}\right)I_0\left(\frac{xv}{{\sigma }^2}\right)$$

Percent Point Function / Sample

$${\text{Sample}}_X=\sqrt{{\mathrm{\Phi }}^{-1}(u_1,v,\sigma {)}^2+{\mathrm{\Phi }}^{-1}(u_2,0,\sigma {)}^2}$$

Parametric Centered Moments

$${\mu }_k^{\prime }=E[X^k]={\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}{x}^k{f}_X\left(x\right)dx={\sigma }^k{2}^{k/2}\mathrm{\Gamma }(1+k/2)L_{k/2}(-v^2/2{\sigma }^2)$$

Parametric Mean

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

Parametric Variance

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

Parametric Skewness

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

Parametric Kurtosis

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

Parametric Median

$$\mathrm{Median}(X)=F_X^{-1}\left(\frac{1}{2}\right)$$

Parametric Mode

$$\mathrm{Mode}(X)=\arg \underset{x}{max}{f}_X\left(x\right)$$

Additional Information and Definitions

• $$\begin{gathered} \text{Computing an analytic expression for the inverse of the cumulative distribution function is not} \\ \text{feasible. Nonetheless, it is possible to generate a random sample from the distribution.} \end{gathered}$$
• ${\mathrm{\Phi }}^{-1}(u,mean,variance):\text{Inverse of cumulative function from normal distribution}$
• $L_r\left(x\right):\text{Laguerre polynomials of order }r\in \mathbb{R}$
• $L_{\frac{1}{2}}\left(x\right)=e^{x/2}(x)I_1\left(\frac{x}{2}\right)-e^{x/2}(x-1)I_0\left(\frac{x}{2}\right)$
• $L_{\frac{3}{2}}\left(x\right)=\frac{1}{3}{e}^{x/2}(2x^2-6x+3)I_0(x/2)-\frac{2}{3}{e}^{x/2}(x-2)x{I}_1(x/2)$
• $I_{\alpha }\left(x\right):\text{Modified Bessel function of the first kind of order }\alpha \in \mathbb{N}$
• $Q_k(a,b):\text{Marcum Q-function of order k }\in \mathbb{N}$
• $u_1:\text{Uniform[0,1] random varible}$
• $u_2:\text{Uniform[0,1] random varible}$

Spreadsheet Documents

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