The Beta distribution

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The beta distribution describes a distribution in which outcomes are limited to a specific range, the probability density function within this range being characterised by two shape parameters.




Distribution name

Beta distribution

Common notation


 = shape parameter (

 = shape parameter (


Probability density function

Cumulative distribution function




(Excess) kurtosis

Characteristic function

Other comments

The beta distribution is also known as a beta distribution of the first kind. Its mode is . There is no simple closed form solution for its median.


The beta distribution parameters are sometimes taken to include boundary parameters  () in which case its domain is , and its pdf and cdf are  and  where , its mean is , its mode is  for  and its variance is  


If  and  then . If  then  and if  then  (if  and ). The ’th order statistic of a sample of size  from the uniform distribution has a beta distribution, .


If  then , i.e. the Pert distribution is a special case of the beta distribution.


Its non-central moments are .


Nematrian web functions


Functions relating to the above distribution may be accessed via the Nematrian web function library by using a DistributionName of “beta”. Functions relating to a generalised version of this distribution including additional location (i.e. shift) and scale parameters may be accessed by using a DistributionName of “beta4”, see also including additional shift and scale parameters. For details of other supported probability distributions see here.


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