PERT DISTRIBUTION

Phitter implementation

Distribution Definition

import phitter

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

💡 The distribution's parameters are defined equation section below

Distribution Methods and Attributes

## 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

$$X\sim \mathrm{Pert}(a,b,c)$$

Distribution Domain

$$x\in [a,c]$$

Parameters Domain and Constraints

$$a\in \mathbb{R},b\in \mathbb{R},c\in \mathbb{R},a<b<c$$

Cumulative Distribution Function

$$F_X\left(x\right)=I(z(x),{\alpha }_1,{\alpha }_2)$$

Probability Density Function

$$f_X\left(x\right)=\frac{(x-a{)}^{{\alpha }_1-1}(c-x{)}^{{\alpha }_2-1}}{\text{Beta}({\alpha }_1,{\alpha }_2)(c-a{)}^{{\alpha }_1+{\alpha }_2-1}}$$

Percent Point Function / Sample

$$F_X^{-1}\left(u\right)=a+(c-a)\cdot I^{-1}(u,{\alpha }_1,{\alpha }_2)$$

Parametric Centered Moments

$${\mu }_k^{\prime }=E[X^k]={\int }_a^c{x}^k{f}_X\left(x\right)dx$$

Parametric Mean

$$\mathrm{Mean}(X)={\mu }_1^{\prime }=\frac{a+4b+c}{6}$$

Parametric Variance

$$\mathrm{Variance}(X)={\mu }_2^{\prime }-{\mu }_1^{\prime 2}=\frac{(\mathrm{Mean}(X)-a)(c-\mathrm{Mean}(X))}{7}$$

Parametric Skewness

$$\mathrm{Skewness}(X)=\frac{{\mu }_3^{\prime }-3{\mu }_2^{\prime }{\mu }_1^{\prime }+2{\mu }_1^{\prime 3}}{({\mu }_2^{\prime }-{\mu }_1^{\prime 2}{)}^{1.5}}=\frac{2({\alpha }_2-{\alpha }_1)\sqrt{{\alpha }_1+{\alpha }_2+1}}{({\alpha }_1+{\alpha }_2+2)\sqrt{{\alpha }_1{\alpha }_2}}$$

Parametric Kurtosis

$$\mathrm{Kurtosis}(X)=\frac{{\mu }_4^{\prime }-4{\mu }_1^{\prime }{\mu }_3^{\prime }+6{\mu }_1^{\prime 2}{\mu }_2^{\prime }-3{\mu }_1^{\prime 4}}{({\mu }_2^{\prime }-{\mu }_1^{\prime 2}{)}^2}=\frac{6[({\alpha }_1-{\alpha }_2{)}^2({\alpha }_1+{\alpha }_2+1)-{\alpha }_1{\alpha }_2({\alpha }_1+{\alpha }_2+2)]}{{\alpha }_1{\alpha }_2({\alpha }_1+{\alpha }_2+2)({\alpha }_1+{\alpha }_2+3)}+3$$

Parametric Median

$$\mathrm{Median}(X)=a+(c-a)\cdot I^{-1}(\frac{1}{2},{\alpha }_1,{\alpha }_2)$$

Parametric Mode

$$\mathrm{Mode}(X)=b$$

Additional Information and Definitions

• $z\left(x\right)=(x-a)/(c-a)$
• $u:\text{Uniform[0,1] random varible}$
• ${\alpha }_1=\frac{4b+c-5a}{c-a},{\alpha }_2=\frac{5c-a-4b}{c-a}$
• $I(x,a,b):\text{Regularized incomplete beta function}$
• $I^{-1}(x,a,b):\text{Inverse of regularized incomplete beta function}$
• $\text{Beta}(x,y):\text{Beta function}$

Spreadsheet Documents

• Phitter playground
• Download Excel Spreadsheet
• Excel file from GitHub repository
• Google spreadsheet document