In probability theory

Probability theory

Probability theory is the branch of mathematics concerned with analysis of random phenomena. The central objects of probability theory are random variables, stochastic processes, and events: mathematical abstractions of non-deterministic events or measured quantities that may either be single...

 and statistics

Statistics

Statistics is the study of the collection, organization, analysis, and interpretation of data. It deals with all aspects of this, including the planning of data collection in terms of the design of surveys and experiments....

, the beta distribution is a family of continuous probability distribution

Probability distribution

In probability theory, a probability mass, probability density, or probability distribution is a function that describes the probability of a random variable taking certain values....

s defined on the interval (0, 1) parameterized by two positive shape parameter

Shape parameter

In probability theory and statistics, a shape parameter is a kind of numerical parameter of a parametric family of probability distributions.- Definition :...

s, typically denoted by α and β. The beta distribution can be suited to the statistical modelling of proportions in applications where values of proportions equal to 0 or 1 do not occur. One theoretical case where the beta distribution arises is as the distribution of the ratio formed by one random variable

Random variable

In probability and statistics, a random variable or stochastic variable is, roughly speaking, a variable whose value results from a measurement on some type of random process. Formally, it is a function from a probability space, typically to the real numbers, which is measurable functionmeasurable...

 having a Gamma distribution divided by the sum of it and another independent

Statistical independence

In probability theory, to say that two events are independent intuitively means that the occurrence of one event makes it neither more nor less probable that the other occurs...

 random variable also having a Gamma distribution with the same scale parameter (but possibly different shape parameter).

The usual formulation of the beta distribution is also known as the beta distribution of the first kind, while beta distribution of the second kind has the alternative name of the beta prime distribution.
The probability density function

Probability density function

In probability theory, a probability density function , or density of a continuous random variable is a function that describes the relative likelihood for this random variable to occur at a given point. The probability for the random variable to fall within a particular region is given by the...

 of the beta distribution is:


where is the gamma function

Gamma function

In mathematics, the gamma function is an extension of the factorial function, with its argument shifted down by 1, to real and complex numbers...

.