PERT distribution

PERT
	Probability density function Example density curves for the PERT probability distribution
	Cumulative distribution function Example cumulative distribution curves for the PERT probability distribution
Parameters	(real) ; (real)
Support
PDF	where
CDF	(the regularized incomplete beta function) with
Mean
Median
Mode
Variance
Skewness
Ex. kurtosis

In probability and statistics, the PERT distribution is a family of continuous probability distributions defined by the minimum (a), most likely (b) and maximum (c) values that a variable can take. It is a transformation of the four-parameter Beta distribution with an additional assumption that its expected value is

\mu ={\frac {a+4b+c}{6}}.

The mean of the distribution is therefore defined as the weighted average of the minimum, most likely and maximum values that the variable may take, with four times the weight applied to the most likely value. This assumption about the mean was first proposed in Clark, 1962[1] for estimating the effect of uncertainty of task durations on the outcome of a project schedule being evaluated using the program evaluation and review technique, hence its name. The mathematics of the distribution resulted from the authors' desire to make the standard deviation equal to about 1/6th of the range.[2][3] The PERT distribution is widely used in risk analysis[4] to represent the uncertainty of the value of some quantity where one is relying on subjective estimates, because the three parameters defining the distribution are intuitive to the estimator. The PERT distribution is featured in most simulation software tools.

Comparison with the triangular distribution

Comparing density curves for the PERT and triangular probability distributions

The PERT distribution offers an alternative[5] to using the triangular distribution which takes the same three parameters. The PERT distribution has a smoother shape than the Triangular distribution. The triangular distribution has a mean equal to the average of the three parameters:

\mu ={\frac {a+b+c}{3}}

The formula places equal emphasis on the extreme values which are usually less-well known than the most likely value, and can therefore be unduly influenced by poor estimation of an extreme. The triangular distribution also has an angular shape that does not match the smoother shape that typifies subjective knowledge:

The modified-PERT distribution

Comparing density curves for the modified PERT distributions with different weights

The PERT distribution assigns very small probability to extreme values, particularly to the extreme furthest away from the most likely value if the distribution is strongly skewed.[6][7] The Modified PERT distribution [8] was proposed to provide more control on how much probability is assigned to tail values of the distribution. The modified-PERT introduces a fourth parameter $\gamma ,$ that controls the weight of the most likely value in the determination of the mean:

\mu ={\frac {a+\gamma b+c}{\gamma +2}}

Typically, values of between 2 and 3.5 are used for $\gamma ,$ and have the effect of flattening the density curve. This is useful for highly skewed distributions where the distances $(b-a),$ and $(c-b),$ are of very different sizes.

The modified-PERT distribution has been implemented in several simulation packages and programming languages:

ModelRisk[9] – risk analysis add-in for Excel.
Primavera risk analysis – project risk analysis simulation tool.
Tamara [10] – project risk analysis simulation tool.
Wolfram Mathematica[11] – mathematical symbolic computation program.
R (programming language): mc2d package.[12]
Python (programming language): pertdist package.[13]

References

Clark CE (1962) The PERT model for the distribution of an activity. Operations Research 10, pp. 405406
"PERT distribution". Vose Software. 2017-05-02. Retrieved 2017-07-16.
Continuous Univariate Distributions - 2nd Ed (1995). Johnson K, Kotz S and Balakkrishnan N. (Section 25.4)
Project Management Body of Knowledge: 5th Ed (2013). Project Management Institute Chapter 6
Simulation Modeling and Analysis (2000). Law AM and Kelton WD. Section 6.11
Business Risk and Simulation Modelling in Practice (2015). M Rees. Section 9.1.8
Risk Analysis – a Quantitative Guide: 3rd Ed. (2008) Vose D
Paulo Buchsbaum (June 9, 2012). "Modified Pert Simulation" (PDF). Greatsolutions.com.br. Archived from the original on December 23, 2018. Retrieved July 14, 2017.
"Modified PERT distribution". Vose Software. 2017-05-02. Retrieved 2017-07-16.
"Probability distributions used in Tamara". Vose Software. 2017-05-02. Retrieved 2017-07-16.
"PERTDistribution—Wolfram Language Documentation". Reference.wolfram.com. Retrieved 2017-07-16.
"Package 'mc2d'" (PDF). 2017-03-06. Retrieved 2020-12-16.
"PertDist". 2020-12-06. Retrieved 2021-01-05.

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[1] Clark CE (1962) The PERT model for the distribution of an activity. Operations Research 10, pp. 405406

[2] "PERT distribution". Vose Software. 2017-05-02. Retrieved 2017-07-16.

[3] Continuous Univariate Distributions - 2nd Ed (1995). Johnson K, Kotz S and Balakkrishnan N. (Section 25.4)

[4] Project Management Body of Knowledge: 5th Ed (2013). Project Management Institute Chapter 6

[5] Simulation Modeling and Analysis (2000). Law AM and Kelton WD. Section 6.11

[6] Business Risk and Simulation Modelling in Practice (2015). M Rees. Section 9.1.8

[7] Risk Analysis – a Quantitative Guide: 3rd Ed. (2008) Vose D

[8] Paulo Buchsbaum (June 9, 2012). "Modified Pert Simulation" (PDF). Greatsolutions.com.br. Archived from the original on December 23, 2018. Retrieved July 14, 2017.

[9] "Modified PERT distribution". Vose Software. 2017-05-02. Retrieved 2017-07-16.

[10] "Probability distributions used in Tamara". Vose Software. 2017-05-02. Retrieved 2017-07-16.

[11] "PERTDistribution—Wolfram Language Documentation". Reference.wolfram.com. Retrieved 2017-07-16.

[12] "Package 'mc2d'" (PDF). 2017-03-06. Retrieved 2020-12-16.

[13] "PertDist". 2020-12-06. Retrieved 2021-01-05.

Probability distributions (List)
Discrete univariate with finite support	Benford Bernoulli beta-binomial binomial categorical hypergeometric Poisson binomial Rademacher soliton discrete uniform Zipf Zipf–Mandelbrot
Discrete univariate with infinite support	beta negative binomial Borel Conway–Maxwell–Poisson discrete phase-type Delaporte extended negative binomial Flory–Schulz Gauss–Kuzmin geometric logarithmic negative binomial Panjer parabolic fractal Poisson Skellam Yule–Simon zeta
Continuous univariate supported on a bounded interval	arcsine ARGUS Balding–Nichols Bates beta beta rectangular continuous Bernoulli Irwin–Hall Kumaraswamy logit-normal noncentral beta PERT raised cosine reciprocal triangular U-quadratic uniform Wigner semicircle
Continuous univariate supported on a semi-infinite interval	Benini Benktander 1st kind Benktander 2nd kind beta prime Burr chi-squared chi Dagum Davis exponential-logarithmic Erlang exponential F folded normal Fréchet gamma gamma/Gompertz generalized gamma generalized inverse Gaussian Gompertz half-logistic half-normal Hotelling's T-squared hyper-Erlang hyperexponential hypoexponential inverse chi-squared scaled inverse chi-squared inverse Gaussian inverse gamma Kolmogorov Lévy log-Cauchy log-Laplace log-logistic log-normal Lomax matrix-exponential Maxwell–Boltzmann Maxwell–Jüttner Mittag-Leffler Nakagami noncentral chi-squared noncentral F Pareto phase-type poly-Weibull Rayleigh relativistic Breit–Wigner Rice shifted Gompertz truncated normal type-2 Gumbel Weibull discrete Weibull Wilks's lambda
Continuous univariate supported on the whole real line	Cauchy exponential power Fisher's z Gaussian q generalized normal generalized hyperbolic geometric stable Gumbel Holtsmark hyperbolic secant Johnson's S_U Landau Laplace asymmetric Laplace logistic noncentral t normal (Gaussian) normal-inverse Gaussian skew normal slash stable Student's t type-1 Gumbel Tracy–Widom variance-gamma Voigt
Continuous univariate with support whose type varies	generalized chi-squared generalized extreme value generalized Pareto Marchenko–Pastur q-exponential q-Gaussian q-Weibull shifted log-logistic Tukey lambda
Mixed continuous-discrete univariate	rectified Gaussian
Multivariate (joint)	Discrete Ewens multinomial Dirichlet-multinomial negative multinomial Continuous Dirichlet generalized Dirichlet multivariate Laplace multivariate normal multivariate stable multivariate t normal-inverse-gamma normal-gamma Matrix-valued inverse matrix gamma inverse-Wishart matrix normal matrix t matrix gamma normal-inverse-Wishart normal-Wishart Wishart
Directional	Univariate (circular) directional Circular uniform univariate von Mises wrapped normal wrapped Cauchy wrapped exponential wrapped asymmetric Laplace wrapped Lévy Bivariate (spherical) Kent Bivariate (toroidal) bivariate von Mises Multivariate von Mises–Fisher Bingham
Degenerate and singular	Degenerate Dirac delta function Singular Cantor
Families	Circular compound Poisson elliptical exponential natural exponential location–scale maximum entropy mixture Pearson Tweedie wrapped

PERT
Probability density function Example density curves for the PERT probability distribution
Cumulative distribution function Example cumulative distribution curves for the PERT probability distribution
Parameters	$b>a\,$ (real) $c>b\,$ (real)
Support	$x\in [a,c]\,$
PDF	${\frac {(x-a)^{\alpha -1}(c-x)^{\beta -1}}{\mathrm {B} (\alpha ,\beta )(c-a)^{\alpha +\beta -1}}}$ where $\alpha ={\frac {4b+c-5a}{c-a}}=1+4{\frac {b-a}{c-a}}$ $\beta ={\frac {5c-a-4b}{c-a}}=1+4{\frac {c-b}{c-a}}$
CDF	$I_{z}(\alpha ,\beta )$ (the regularized incomplete beta function) with $z=(x-a)/(c-a)$
Mean	$\operatorname {E} [X]={\frac {a+4b+c}{6}}=\mu$
Median	$I_{\frac {1}{2}}^{[-1]}(\alpha ,\beta )(c-a)+a$ $\approx a+(c-a){\frac {\alpha -1/3}{\alpha +\beta -2/3}}={\frac {a+6b+c}{8}}$
Mode	$b$
Variance	$\operatorname {var} [X]={\frac {(\mu -a)(c-\mu )}{7}}$
Skewness	${\frac {2\,(\beta -\alpha ){\sqrt {\alpha +\beta +1}}}{(\alpha +\beta +2){\sqrt {\alpha \beta }}}}$
Ex. kurtosis	${\frac {6[(\alpha -\beta )^{2}(\alpha +\beta +1)-\alpha \beta (\alpha +\beta +2)]}{\alpha \beta (\alpha +\beta +2)(\alpha +\beta +3)}}$