UNIFORM DISTRIBUTION

Phitter implementation

Distribution Definition

python

import phitter

distribution = phitter.continuous.Uniform({"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 Uniform (a, b)

Distribution Domain

x \in [a, b]

Parameters Domain and Constraints

a \in R, b \in R, a < b

Cumulative Distribution Function

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

Probability Density Function

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

Percent Point Function / Sample

F_{X}^{- 1} (u) = a + u \cdot (b - a)

Parametric Centered Moments

μ_{k}^{'} = E [X^{k}] = \int_{- \infty}^{\infty} x^{k} f_{X} (x) d x = \frac{1}{k + 1} \sum_{i = 0}^{k} a^{i} b^{k - i}

Parametric Mean

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

Parametric Variance

Variance (X) = μ_{2}^{'} - μ_{1}^{' 2}

Parametric Skewness

Skewness (X) = \frac{μ_{3}^{'} - 3 μ_{2}^{'} μ_{1}^{'} + 2 μ_{1}^{' 3}}{(μ_{2}^{'} - μ_{1}^{' 2})^{1.5}} = 0

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}} = 3 - \frac{6}{5}

Parametric Median

Median (X) = \frac{1}{2} (a + b)

Parametric Mode

Mode (X) \in [a, b]

Additional Information and Definitions

$u : Uniform[0,1] random varible$

Spreadsheet Documents

UNIFORM DISTRIBUTION ​

Phitter implementation ​

Equations ​

UNIFORM DISTRIBUTION

Phitter implementation

Equations