Beta Distribution

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The beta distribution (also called the beta distribution of the first kind) is a family of continuous probability distributions defined on [0, 1]. The beta distribution is similar to the binomial distribution, except where the binomial models the number of successes (x), the beta models the probability (p) of success.

Beta distribution of the second kind is another name for the beta prime distribution.

Beta Distribution PDF and CDF

The beta distribution is parameterized by two free parameters, alpha (α) and beta (β), which control the shape of the graph. The gives the notation for the beta distribution, Β (α, β), where α and β are real numbers. Other notation for the shape parameters includes (p, q) [1].

The probability density function (PDF) is


Γ is the gamma function
  • 1/B(α,β) is a normalizing constant to force the function integrate to 1.

The graph of the beta distribution density function can take on a variety of shapes.  For example, if α < 1 and Β < 1, the graph will be in the shape of a “U”, and if α = 1 and Β = 2, the graph is a straight line.

The cumulative distribution function (CDF) is the regularized incomplete beta function

Uses of the Beta Distribution

The beta distribution is used for a variety of applications for modeling the behavior of random variables limited to intervals of finite length. Uses include:

  • Bayesian inference, as the conjugate prior probability distribution for the Bernoulli, binomial, geometric  and negative binomial distributions.
  • The Rule of Succession (such as Pierre-Simon Laplace’s treatment of the sunrise problem),
  • Task duration modeling.
  • Project/planning control systems like PERT and CPM.
  • In project management for the “three-point technique,” (also called the beta distribution technique), which recognizes uncertainty in estimated project time [1]. 


[1] Haight, F. (1958). Index to the Distributions of Mathematical Statistics. National Bureau of Standards Report.

[2] 3-Points Estimating