POWER FUNCTION DISTRIBUTION

Phitter implementation

Distribution Definition

python

import phitter

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

💡 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 PowerFunction (α, a, b)

Distribution Domain

x \in [a, b]

Parameters Domain and Constraints

α \in R^{+}, a \in R, b \in R, a < b

Cumulative Distribution Function

F_{X} (x) = {(\frac{x - a}{b - a})}^{α}

Probability Density Function

f_{X} (x) = \frac{α (x - a)^{α - 1}}{(b - a)^{α}}

Percent Point Function / Sample

F_{X}^{- 1} (u) = {[a + u (b - a)]}^{- α}

Parametric Centered Moments

μ_{k}^{'} = E [X^{k}] = \int_{a}^{b} x^{k} f_{X} (x) d x

Parametric Mean

Mean (X) = μ_{1}^{'} = \frac{a + b α}{α + 1}

Parametric Variance

Variance (X) = μ_{2}^{'} - μ_{1}^{' 2} = \frac{2 a^{2} + 2 a b α + b^{2} α (α + 1)}{(α + 1) (α + 2)} - Mean (X)^{2}

Parametric Skewness

Skewness (X) = \frac{μ_{3}^{'} - 3 μ_{2}^{'} μ_{1}^{'} + 2 μ_{1}^{' 3}}{(μ_{2}^{'} - μ_{1}^{' 2})^{1.5}} = 2 (1 - α) \sqrt{\frac{α + 2}{α (α + 3)}}

Parametric Kurtosis

Kurtosis (X) = \frac{μ_{4}^{'} - 4 μ_{1}^{'} μ_{3}^{'} + 6 μ_{1}^{' 2} μ_{2}^{'} - 3 μ_{1}^{' 4}}{(μ_{2}^{'} - μ_{1}^{' 2})^{2}} = \frac{6 (α^{3} - α^{2} - 6 α + 2)}{α (α + 3) (α + 4)} + 3

Parametric Median

Median (X) = {[a + \frac{1}{2} (b - a)]}^{- α}

Parametric Mode

Mode (X) = undefined

Additional Information and Definitions

$a : Location parameter$
$b - a : Scale parameter$
$u : Uniform[0,1] random varible$

Spreadsheet Documents

POWER FUNCTION DISTRIBUTION ​

Phitter implementation ​

Equations ​

POWER FUNCTION DISTRIBUTION

Phitter implementation

Equations