HYPERBOLIC SECANT DISTRIBUTION

Phitter implementation

Distribution Definition

import phitter

distribution = phitter.continuous.HyperbolicSecant({"mu": *, "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

$$X\sim \mathrm{HyperbolicSecant}(\mu ,\sigma )$$

Distribution Domain

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

Parameters Domain and Constraints

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

Cumulative Distribution Function

$$F\mathrm{\_}X\left(x\right)=\frac{2}{\pi }\arctan \left[\exp (\frac{\pi }{2}z(x))\right]$$

Probability Density Function

$$f\mathrm{\_}X\left(x\right)=\frac{1}{2\sigma }\mathrm{sech}(\frac{\pi }{2}z(x))$$

Percent Point Function / Sample

$$F^{-1}\mathrm{\_}X\left(u\right)=\mu +\sigma \frac{2}{\pi }\ln [\tan \left(\frac{\pi }{2}u\right)]$$

Parametric Centered Moments

$${\tilde{\mu }}_k^{\prime }=E[{\tilde{X}}^k]={\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}{x}^{k}f\mathrm{\_}\tilde{X}\left(x\right)dx=\frac{1+{(-1)}^k}{2\pi 2^{2k}}k![\zeta (k+1,\frac{1}{4})-\zeta (k+1,\frac{3}{4})]$$

Parametric Mean

$$\mathrm{Mean}(X)=\mu +\sigma {\tilde{\mu }}^{\prime }\mathrm{\_}1=\mu$$

Parametric Variance

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

Parametric Skewness

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

Parametric Kurtosis

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

Parametric Median

$$\mathrm{Median}(X)=\mu$$

Parametric Mode

$$\mathrm{Mode}(X)=\mu$$

Additional Information and Definitions

• $\tilde{X}\sim \mathrm{HyperbolicSecant}(0,1)$
• $\mu :\text{Location parameter}$
• $\sigma :\text{Scale parameter}$
• $z\left(x\right)=(x-\mu )/\sigma$
• $u:\text{Uniform[0,1] random varible}$
• $\zeta (a,s):\text{Hurwitz zeta function}$

Spreadsheet Documents

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