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UNIFORM DISTRIBUTION ​

Phitter implementation ​

Distribution Definition

python
import phitter

distribution = phitter.discrete.Uniform({"min": *, "max": *})

💡 The distribution's parameters are defined equation section below

Distribution Methods and Attributes

python
## CDF, PMF, PPF receive float or numpy.ndarray.
distribution.cdf(int | numpy.ndarray[int]) # -> float | numpy.ndarray
distribution.pmf(int | numpy.ndarray[int]) # -> float | numpy.ndarray
distribution.ppf(int | numpy.ndarray[int]) # -> 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 # -> int
distribution.mode # -> int

Equations ​

Distribution Definition

X∼Uniform(a,b)

Distribution Domain

x∈{a,a+1,…,b−1,b}

Parameters Domain and Constraints

a∈N,b∈N,a<b

Cumulative Distribution Function

FX(x)=x−a+1b−a+1

Probability Density Function

fX(x)=1b−a+1

Percent Point Function / Sample

FX−1(u)=⌈u(b−a+1)+a−1⌉

Parametric Centered Moments

E[Xk]=μk′=∑x=abxkfX(x)=1b−a+1∑x=abxk

Parametric Mean

Mean(X)=μ1′=a+b2

Parametric Variance

Variance(X)=(μ2′−μ1′2)=(b−a+1)2−112

Parametric Skewness

Skewness(X)=μ3′−3μ2′μ1′+2μ1′3(μ2′−μ1′2)1.5=0

Parametric Kurtosis

Kurtosis(X)=μ4′−4μ1′μ3′+6μ1′2μ2′−3μ1′4(μ2′−μ1′2)2=3−6((b−a+1)2+1)5((b−a+1)2−1)

Parametric Median

Median(X)=a+b2

Parametric Mode

Mode(X)∈[a,b]

Additional Information and Definitions

  • u:Uniform[0,1] random varible
  • ⌈x⌉:Ceiling Function

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