Beta distribution

Beta Distribution is a family of continuous probability distributions defined on the interval (0, 1) parameterized by two positive shape parameters, denoted by α and β, that appear as exponents of the random variable and control the shape of the distribution.

Definition

The probability density function (pdf) of the beta distribution, for 0 < x < 1, and shape parameters α, β > 0, is a power function of the variable x and of its reflection (1 − x) as follows:

f(x; α, β) = constant × x^(α − 1) × (1 − x)^(β − 1)

The constant is a normalizing constant that makes the total probability equal to one.

Properties

The beta distribution has several unique properties:

• The beta distribution is a suitable model for the random behavior of percentages and proportions.
• The expected value and variance of a random variable following a beta distribution can be calculated using the shape parameters α and β.
• The beta distribution is a conjugate prior probability distribution for the Bernoulli, binomial, negative binomial and geometric distributions.

Applications

The beta distribution has been applied in various fields:

• In project management, the beta distribution is used in PERT to model events which have a bounded range and where the outcomes are most likely to be near the middle than the extremes.
• In machine learning, the beta distribution is used in Bayesian inference.

See also

• Probability distribution
• Continuous distribution
• Conjugate prior
• Bayesian inference