BATES DISTRIBUTION

Phitter implementation

Distribution Definition

python

import phitter

distribution = phitter.continuous.Bates({"n": *, "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 Bates (n, min, max)

Distribution Domain

x \in (min, max)

Parameters Domain and Constraints

n \in N^{+}, min \in R, max \in R

Cumulative Distribution Function

F_{X} (x) = \frac{1}{n!} \sum_{k = 0}^{⌊ y ⌋} {(- 1)}^{k} (\binom{n}{k}) {(y - k)}^{n}

Probability Density Function

f_{X} (x) = \frac{n}{(max - min) (n - 1)!} \sum_{k = 0}^{⌊ y ⌋} {(- 1)}^{k} (\binom{n}{k}) {(y - k)}^{n - 1}

Percent Point Function / Sample

F_{X}^{- 1} (u) = numerical inversion of F_{X}

Parametric Centered Moments

{\tilde{μ}}_{k} = E [{(Y - \frac{1}{2})}^{k}]

Parametric Mean

Mean (X) = \frac{min + max}{2}

Parametric Variance

Variance (X) = \frac{{(max - min)}^{2}}{12 n}

Parametric Skewness

Skewness (X) = 0

Parametric Kurtosis

Kurtosis (X) = 3 - \frac{6}{5 n}

Parametric Median

Median (X) = \frac{min + max}{2}

Parametric Mode

Mode (X) = \frac{min + max}{2}

Additional Information and Definitions

$Computing an analytic expression for the inverse of the cumulative distribution function is not$ $feasible. Nonetheless, it is possible to generate a random sample from the distribution.$
$X = min + (max - min) \cdot \frac{1}{n} \sum_{i = 1}^{n} U_{i}$
$y = n \cdot (x - min) / (max - min)$
$n : Number of averaged uniforms (Irwin-Hall order)$
$min : Lower bound of the support$
$max : Upper bound of the support$
$(\binom{n}{k}) : Binomial coefficient$
$U_{i} : Uniform[0,1] random varible$

Spreadsheet Documents

BATES DISTRIBUTION ​

Phitter implementation ​

Equations ​

BATES DISTRIBUTION

Phitter implementation

Equations