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TRIANGULAR DISTRIBUTION ​

Phitter implementation ​

Distribution Definition

python
import phitter

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

💡 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∼Triangular(a,b,c)

Distribution Domain

x∈[a,b]

Parameters Domain and Constraints

a∈R,b∈R,c∈R,a<c<b

Cumulative Distribution Function

FX(x)={(x−a)2(b−a)(c−a)if  a<x≤c1−(b−x)2(b−a)(b−c)if  c<x<b

Probability Density Function

fX(x)={2(x−a)(b−a)(c−a)if  a≤x<c,2(b−x)(b−a)(b−c)if  c≤x≤b,

Percent Point Function / Sample

FX−1(u)={a+U(b−a)(c−a)if  0<U<c−ab−ab−(1−U)(b−a)(b−c)if  c−ab−a≤U<1

Parametric Centered Moments

μk′=E[Xk]=∫abxkfX(x)dx

Parametric Mean

Mean(X)=μ1′=a+b+c3

Parametric Variance

Variance(X)=μ2′−μ1′2=a2+b2+c2−ab−ac−bc18

Parametric Skewness

Skewness(X)=μ3′−3μ2′μ1′+2μ1′3(μ2′−μ1′2)1.5=2(a+b−2c)(2a−b−c)(a−2b+c)5(a2+b2+c2−ab−ac−bc)32

Parametric Kurtosis

Kurtosis(X)=μ4′−4μ1′μ3′+6μ1′2μ2′−3μ1′4(μ2′−μ1′2)2=3−35

Parametric Median

Median(X)={a+(b−a)(c−a)2if c≥a+b2b−(b−a)(b−c)2if c≤a+b2

Parametric Mode

Mode(X)∈[b,c]

Additional Information and Definitions

  • u:Uniform[0,1] random varible

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