Four-parameter Beta distribution

Format: Beta(a, b, min, max)


The four-parameter beta distribution is highly flexible in shape and bounded, so has been quite popular for attempting to fit to a data set for a bounded variable. In ModelRisk we offer the option of fitting the four-parameter beta distribution with known bounds (our general recommendation) or without.

The PERT distribution came out of the need to describe the uncertainty in tasks during the development of the Polaris missile (Clark, 1962). The project had thousands of tasks and estimates needed to be made that were intuitive, quick and consistent in approach. The 4-parameter beta distribution was used just because it came to the author's mind (the Kumaraswamy distribution would also have been a good candidate, for example). The decision to constrain the distribution so that it's Mean = (Min + 4* Mode + Max)/6 was an approximation to their decision that the distribution should have a standard deviation of 1/6 of its range (i.e. Max - Min).

Golenko-Ginzburg (1988) describes a study that analyzed many PERT networks and concluded that 'the 'most likely' activity-time estimate m [mode] is practically useless'. They found that the location of the mode in most project tasks was approximately one third of the distance from the Min to the Max, i.e:

Mode = Min + (Max-Min)/3

Taking the Beta4(a, b, min, max) distribution again, this equates to a = 2,  b = 3.  Thus, from Golenko-Ginzburg's viewpoint it is sufficient to use

Beta4(2, 3, min, max)

in place of

PERT(min, mode, max)

with the added advantage that one is only asking a subject matter expert for two values.

ModelRisk functions added to Microsoft Excel for the Four-Parameter Beta distribution

VoseBeta4 generates random values from this distribution for Monte Carlo simulation, or calculates a percentile if used with a U parameter.

VoseBeta4Object constructs a distribution object for this distribution.

VoseBeta4Prob returns the probability density or cumulative distribution function for this distribution.

VoseBeta4Prob10 returns the log10 of the probability density or cumulative distribution function.

VoseBeta4Fit generates values from this distribution fitted to data, or calculates a percentile from the fitted distribution.

VoseBeta4FitObject constructs a distribution object of this distribution fitted to data.

VoseBeta4FitP returns the parameters of this distribution fitted to data.


Four-parameter Beta distribution equations



Monte Carlo simulation in Excel. Learn more


Adding risk and uncertainty to your project schedule. Learn more



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