beta distribution

Noun

 * 1)  Any of a family of continuous probability distributions defined on the interval [0, 1] whose shape is parametrised by two positive parameters, denoted α and β, which appear as exponents in the associated random variable.
 * 2) * 2014, Stephen F. Bush, Smart Grid: Communication-Enabled Intelligence for the Electric Power Grid, Wiley, 2015, Reprint with corrections, page 366,
 * When more than two mutually exclusive events are involved in a frame of reference, the beta distributions form the marginals of a Dirichlet distribution.
 * 1) * 2014, Nikolay Tcholtchev, Ina Schieferdecker, Framework for Ensuring Runtime Stability of Control Loops in Multi-agent Networked Environments, Marina Gavrilova, C. J. Kenneth Tan (editors), Transactions on Computational Science XXII, Springer, LNCS 8360, page 80,
 * We developed a Matlab script that implements the machine learning procedures as specified by (11), as well as by (13) with applying the beta-distribution (14) as a probability measure.
 * We developed a Matlab script that implements the machine learning procedures as specified by (11), as well as by (13) with applying the beta-distribution (14) as a probability measure.

Usage notes
Of importance in as the conjugate prior probability distribution for the Bernoulli, binomial, negative binomial and geometric distributions. (That is, it can be used, among other things, to describe initial knowledge of probability of success.)

Mathematically, the probability density function, for 0 ≤ x ≤ 1, can be expressed as $$\textstyle f(x;\alpha,\beta) = \frac{1}{\Beta(\alpha,\beta)} x^{\alpha-1}(1-x)^{\beta-1}$$, where the beta function B is a normalisation constant that ensures the probability function integrates to 1 over the interval.

Translations

 * Finnish: beta-jakauma, β-jakauma