KUMARASWAMY DISTRIBUTION

Phitter implementation

Distribution Definition

python

import phitter

distribution = phitter.continuous.Kumaraswamy({"alpha": *, "beta": *, "min": *, "max": *})

💡 The distribution's parameters are defined equation section below

Distribution Methods and Attributes

python

## 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 Kumaraswamy (α, β, min, max)

Distribution Domain

x \in (min, max)

Parameters Domain and Constraints

α \in R^{+}, β \in R^{+}, min \in R, max \in R

Cumulative Distribution Function

F_{X} (x) = 1 - (1 - z (x)^{α})^{β}

Probability Density Function

f_{X} (x) = α β z (x)^{α - 1} (1 - z (x)^{α})^{β - 1}

Percent Point Function / Sample

F_{X}^{- 1} (u) = min + (max - min) \times (1 - (1 - u)^{\frac{1}{β}})^{\frac{1}{α}}

Parametric Centered Moments

{\tilde{μ}}_{k}^{'} = E [{\tilde{X}}^{k}] = \int_{0}^{1} x^{k} f_{\tilde{X}} (x) d x = β Beta (1 + \frac{k}{α}, β)

Parametric Mean

Mean (X) = min + (max - min) \times {\tilde{μ}}_{1}^{'}

Parametric Variance

Variance (X) = {(max - min)}^{2} ({\tilde{μ}}_{2}^{'} - {\tilde{μ}}_{1}^{' 2})

Parametric Skewness

Skewness (X) = \frac{{\tilde{μ}}_{3}^{'} - 3 {\tilde{μ}}_{2}^{'} {\tilde{μ}}_{1}^{'} + 2 {\tilde{μ}}_{1}^{' 3}}{({\tilde{μ}}_{2}^{'} - {\tilde{μ}}_{1}^{' 2})^{1.5}}

Parametric Kurtosis

Kurtosis (X) = \frac{{\tilde{μ}}_{4}^{'} - 4 {\tilde{μ}}_{1}^{'} {\tilde{μ}}_{3}^{'} + 6 {\tilde{μ}}_{1}^{' 2} {\tilde{μ}}_{2}^{'} - 3 {\tilde{μ}}_{1}^{' 4}}{({\tilde{μ}}_{2}^{'} - {\tilde{μ}}_{1}^{' 2})^{2}}

Parametric Median

Median (X) = min + (max - min) \times {(1 - 2^{- 1 / b})}^{1 / a}

Parametric Mode

Mode (X) = min + (max - min) \times {(\frac{a - 1}{a b - 1})}^{1 / a}

Additional Information and Definitions

$\tilde{X} \sim Kumaraswamy (α, β, 0, 1)$
$z (x) = (x - min) / (max - min)$
$u : Uniform[0,1] random varible$
$Beta (x, y) : Beta function$

Spreadsheet Documents

KUMARASWAMY DISTRIBUTION ​

Phitter implementation ​

Equations ​

KUMARASWAMY DISTRIBUTION

Phitter implementation

Equations