represents a PERT distribution with range min to max and mode at c.


represents a modified PERT distribution with shape parameter λ.


Background & Context


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Basic Examples  (5)

Probability density function:

Cumulative distribution function:




Scope  (8)

Generate a sample of pseudorandom numbers from a PERT distribution:

Compare its histogram to the PDF:

Distribution parameters estimation:

Estimate the distribution parameters from sample data:

Compare the density histogram of the sample with the PDF of the estimated distribution:


For large λ, the modified PERT distribution becomes symmetric:

Limiting values:


For large λ, kurtosis nears the kurtosis of NormalDistribution:

Limiting values:

Different moments with closed forms as functions of parameters:


Closed form for symbolic order:


Closed form for symbolic order:



Hazard function:

Quantile function:

Consistent use of Quantity in parameters yields QuantityDistribution:

Find the quartiles of the project completion time:

Applications  (2)

An expert estimates that a project that takes from 4 to 6 months to complete will take 5 months and 1 week:

The distribution of the project completion time:

Find the expected completion time and its standard deviation:

Find the probability of the project taking longer to complete:

Use PERTDistribution as a smooth alternative to TriangularDistribution:

Properties & Relations  (5)

PERT distribution is closed under translation and scaling by a positive factor:

Relationships to other distributions:

Default value of the shape parameter λ is 4:

PERT distribution is a transformation of BetaDistribution:

BetaDistribution with parameters and is a special case of PERTDistribution on the unit interval:

The equivalent PERT distribution is only valid for and :

Neat Examples  (1)

PDFs for different c values with CDF contours:

Introduced in 2010
Updated in 2016