In
probability
Probability is the branch of mathematics concerning numerical descriptions of how likely an Event (probability theory), event is to occur, or how likely it is that a proposition is true. The probability of an event is a number between 0 and ...
and
statistics
Statistics (from German language, German: ''wikt:Statistik#German, Statistik'', "description of a State (polity), state, a country") is the discipline that concerns the collection, organization, analysis, interpretation, and presentation of ...
, the PERT distribution is a family of
continuous probability distribution
In probability theory and statistics, a probability distribution is the mathematical function that gives the probabilities of occurrence of different possible outcomes for an experiment. It is a mathematical description of a random phenomenon i ...
s 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
In probability theory and statistics, the beta distribution is a family of continuous probability distributions defined on the interval , 1in terms of two positive parameters, denoted by ''alpha'' (''α'') and ''beta'' (''β''), that appear as ...
with an additional assumption that its
expected value
In probability theory, the expected value (also called expectation, expectancy, mathematical expectation, mean, average, or first moment) is a generalization of the weighted average. Informally, the expected value is the arithmetic mean of a l ...
is
:
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 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
The program evaluation and review technique (PERT) is a statistical tool used in project management, which was designed to analyze and represent the tasks involved in completing a given project.
First developed by the United States Navy in ...
, hence its name. The mathematics of the distribution resulted from the authors' desire to make the standard deviation equal to about 1/6 of the range.
The PERT distribution is widely used in
risk analysis 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
![Triangle v PERT distributions](https://upload.wikimedia.org/wikipedia/commons/8/84/Triangle_v_PERT_distributions.jpg)
The PERT distribution offers an alternative to using the
triangular distribution
In probability theory and statistics, the triangular distribution is a continuous probability distribution with lower limit ''a'', upper limit ''b'' and mode ''c'', where ''a'' < ''b'' and ''a'' ≤ ''c'' ≤ ''b''.
...
which takes the same three parameters. The PERT distribution has a smoother shape than the
triangular distribution
In probability theory and statistics, the triangular distribution is a continuous probability distribution with lower limit ''a'', upper limit ''b'' and mode ''c'', where ''a'' < ''b'' and ''a'' ≤ ''c'' ≤ ''b''.
...
. The triangular distribution has a mean equal to the average of the three parameters:
:
which (unlike PERT) places equal emphasis on the extreme values which are usually less-well known than the most likely value, and is therefore less reliable. The triangular distribution also has an angular shape that does not match the smoother shape that typifies subjective knowledge.
The modified-PERT distribution
![Modified PERT example pdfs](https://upload.wikimedia.org/wikipedia/commons/c/cd/Modified_PERT_example_pdfs.jpg)
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. The Modified PERT distribution 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
that controls the weight of the most likely value in the determination of the mean:
:
Typically, values of between 2 and 3.5 are used for
and have the effect of flattening the density curve; the unmodified PERT would use
. This is useful for highly skewed distributions where the distances
and
are of very different sizes.
The modified-PERT distribution has been implemented in several simulation packages and programming languages:
* ModelRisk – risk analysis add-in for
Excel
ExCeL London (an abbreviation for Exhibition Centre London) is an exhibition centre, international convention centre and former hospital in the Custom House area of Newham, East London. It is situated on a site on the northern quay of the ...
.
*
Primavera risk analysis – project risk analysis simulation tool.
* Tamara – project risk analysis simulation tool.
*
Wolfram Mathematica
Wolfram Mathematica is a software system with built-in libraries for several areas of technical computing that allow machine learning, statistics, symbolic computation, data manipulation, network analysis, time series analysis, NLP, optimizat ...
– mathematical symbolic computation program.
*
R (programming language)
R is a programming language for statistical computing and graphics supported by the R Core Team and the R Foundation for Statistical Computing. Created by statisticians Ross Ihaka and Robert Gentleman, R is used among data miners, bioinform ...
: ''mc2d'' package.
*
Python (programming language)
Python is a high-level, general-purpose programming language. Its design philosophy emphasizes code readability with the use of significant indentation.
Python is dynamically-typed and garbage-collected. It supports multiple programming para ...
: ''pertdist'' package.
References
{{ProbDistributions, continuous-bounded
Continuous distributions