ARCSINE DISTRIBUTION

Phitter implementation

Distribution Definition

import phitter

distribution = phitter.continuous.Arcsine({"a": *, "b": *})

💡 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

$$X\sim \mathrm{Arcsine}(a,b)$$

Distribution Domain

$$x\in (a,b)$$

Parameters Domain and Constraints

$$a\in \mathbb{R},b\in \mathbb{R},a<b$$

Cumulative Distribution Function

$$F_X(x)=\frac{2}{\pi }\arcsin \left(\sqrt{\frac{x-a}{b-a}}\right)$$

Probability Density Function

$$f_X(x)=\frac{1}{\pi \sqrt{(x-a)(b-x)}}$$

Percent Point Function / Sample

$$F_X^{-1}\left(u\right)=a+(b-a)\times {\sin }^2\left(\frac{\pi }{2}u\right)$$

Parametric Centered Moments

$${\tilde{\mu }}_k^{\prime }=E[{\tilde{X}}^k]={\int }_0^1{x}^k{f}_{\tilde{X}}\left(x\right)dx=\frac{1}{\pi }\text{Beta}(\frac{1}{2},k+\frac{1}{2})=\frac{(2k-1)!!}{2^{k}k!}$$

Parametric Mean

$$\mathrm{Mean}(X)=a+{\tilde{\mu }}_1^{\prime }(b-a)=a+\frac{1}{2}(b-a)$$

Parametric Variance

$$\mathrm{Variance}(X)={(b-a)}^2\times ({\tilde{\mu }}_2^{\prime }-{\tilde{\mu }}_1^{\prime 2})=\frac{{(b-a)}^2}{8}$$

Parametric Skewness

$$\mathrm{Skewness}(X)=\frac{{\tilde{\mu }}_3^{\prime }-3{\tilde{\mu }}_2^{\prime }{\tilde{\mu }}_1^{\prime }+2{\tilde{\mu }}_1^{\prime 3}}{({\tilde{\mu }}_2^{\prime }-{\tilde{\mu }}_1^{\prime 2}{)}^{1.5}}=0$$

Parametric Kurtosis

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

Parametric Median

$$\mathrm{Median}(X)=a+(b-a)\times {\sin }^2\left(\frac{\pi }{4}\right)$$

Parametric Mode

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

Additional Information and Definitions

• $\tilde{X}\sim \mathrm{Arcsine}(0,1)$
• $u:\text{Uniform[0,1] random varible}$
• $\text{Beta}(x,y):\text{Beta function}$

Spreadsheet Documents

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