BETA DISTRIBUTION

Phitter implementation

Distribution Definition

python

import phitter

distribution = phitter.continuous.Beta({"alpha": *, "beta": *, "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 Beta (α, β, A, B)

Distribution Domain

x \in (A, B)

Parameters Domain and Constraints

α \in R^{+}, β \in R^{+}, A \in R, B \in R, A < B

Cumulative Distribution Function

F_{X} (x) = I (z (x), α, β)

Probability Density Function

f_{X} (x) = \frac{z (x)^{α - 1} {(1 - z (x))}^{β - 1}}{Beta (α, β) (B - A)}

Percent Point Function / Sample

F_{X}^{- 1} (u) = A + (B - A) \times I^{- 1} (u, α, β)

Parametric Centered Moments

{\tilde{μ}}_{k}^{'} = E [{\tilde{X}}^{k}] = \int_{0}^{1} x^{k} f_{\tilde{X}} (x) d x

Parametric Mean

Mean (X) = A + (B - A) \cdot {\tilde{μ}}_{1}^{'} = A + \frac{α (B - A)}{α + β}

Parametric Variance

Variance (X) = {(B - A)}^{2} \cdot ({\tilde{μ}}_{2}^{'} - {\tilde{μ}}_{1}^{' 2}) = \frac{α β {(B - A)}^{2}}{(α + β)^{2} (α + β + 1)}

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}} = \frac{2 (β - α) \sqrt{α + β + 1}}{(α + β + 2) \sqrt{α β}}

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}} = 3 + \frac{6 [(α - β)^{2} (α + β + 1) - α β (α + β + 2)]}{α β (α + β + 2) (α + β + 3)}

Parametric Median

Median (X) = A + (B - A) \times I^{- 1} (\frac{1}{2}, α, β) if α, β > 1

Parametric Mode

Mode (X) = A + (B - A) \frac{α - 1}{α + β - 2} if α, β > 1

Additional Information and Definitions

$\tilde{X} \sim Beta (α, β, 0, 1)$
$z (x) = (x - A) / (B - A)$
$u : Uniform[0,1] random varible$
$I (x, a, b) : Regularized incomplete beta function$
$I^{- 1} (x, a, b) : Inverse of regularized incomplete beta function$
$Beta (x, y) : Beta function$

Spreadsheet Documents

BETA DISTRIBUTION ​

Phitter implementation ​

Equations ​

BETA DISTRIBUTION

Phitter implementation

Equations