BETA PRIME DISTRIBUTION

Phitter implementation

Distribution Definition

python

import phitter

distribution = phitter.continuous.BetaPrime({"alpha": *, "beta": *})

💡 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 BetaPrime (α, β)

Distribution Domain

x \in [0, \infty)

Parameters Domain and Constraints

α \in R^{+}, β \in R^{+}

Cumulative Distribution Function

F_{X} (x) = I (\frac{x}{1 + x}, α, β)

Probability Density Function

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

Percent Point Function / Sample

F_{X}^{- 1} (u) = \frac{I^{- 1} (u, α, β)}{1 - I^{- 1} (u, α, β)}

Parametric Centered Moments

μ_{k}^{'} = E [X^{k}] = \int_{0}^{\infty} x^{k} f_{X} (x) d x = \frac{Γ (k + α) Γ (β - k)}{Γ (α) Γ (β)} if β > k

Parametric Mean

Mean (X) = μ_{1}^{'} = \frac{α}{β - 1} if β > 1

Parametric Variance

Variance (X) = μ_{2}^{'} - μ_{1}^{' 2} = \frac{α (α + β - 1)}{(β - 2) (β - 1)^{2}} if β > 2

Parametric Skewness

Skewness (X) = \frac{μ_{3}^{'} - 3 μ_{2}^{'} μ_{1}^{'} + 2 μ_{1}^{' 3}}{(μ_{2}^{'} - μ_{1}^{' 2})^{1.5}} = \frac{2 (2 α + β - 1)}{β - 3} \sqrt{\frac{β - 2}{α (α + β - 1)}} if β > 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}} if β > 4

Parametric Median

Median (X) = \frac{I^{- 1} (\frac{1}{2}, α, β)}{1 - I^{- 1} (\frac{1}{2}, α, β)}

Parametric Mode

Mode (X) = \frac{α - 1}{β + 1}

Additional Information and Definitions

$u : Uniform[0,1] random varible$
$I (x, a, b) : Regularized incomplete beta function$
$I^{- 1} (x, a, b) : Inverse of regularized incomplete beta function$
$Γ (x) : Gamma function$
$Beta (x, y) : Beta function$

Spreadsheet Documents

BETA PRIME DISTRIBUTION ​

Phitter implementation ​

Equations ​

BETA PRIME DISTRIBUTION

Phitter implementation

Equations