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TRAPEZOIDAL DISTRIBUTION ​

Phitter implementation ​

Distribution Definition

python
import phitter

distribution = phitter.continuous.Trapezoidal({"a": *, "b": *, "c": *, "d": *})

💡 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∼Trapezoidal(a,b,c,d)

Distribution Domain

x∈[a,d]

Parameters Domain and Constraints

a∈R,b∈R,c∈R,d∈R,a<b<c,b<c<d

Cumulative Distribution Function

FX(x)={1d+c−a−b1b−a(x−a)2if  a≤x<b1d+c−a−b(2x−a−b)if  b≤x<c1−1d+c−a−b1d−c(d−x)2if  c≤x≤d

Probability Density Function

fX(x)={2d+c−a−bx−ab−aif  a≤x<b2d+c−a−bif  b≤x<c2d+c−a−bd−xd−cif  c≤x≤d

Percent Point Function / Sample

FX−1(u)={a+u×(d+c−a−b)×(b−a)if u≤A1(a+b+u×(d+c−a−b))/2if A1≤u≤A1+A2d−(1−u)×(d+c−a−b)×(d−c)if A1+A2≤u≤A1+A2+A3

Parametric Centered Moments

μk′=E[Xk]=∫abxkfX(x)dx=2d+c−b−a1(k+1)(k+2)(dk+2−ck+2d−c−bk+2−ak+2b−a)

Parametric Mean

Mean(X)=μ1′

Parametric Variance

Variance(X)=μ2′−μ1′2

Parametric Skewness

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

Parametric Kurtosis

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

Parametric Median

Median(X)=FX−1(1/2)

Parametric Mode

Mode(X)∈[b,c]

Additional Information and Definitions

  • u:Uniform[0,1] random varible
  • A1=(b−a)/(d+c−a−b)
  • A2=2(c−b)/(d+c−a−b)
  • A3=(d−c)/(d+c−a−b)

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