In probability theory and statistics, the beta distribution is a family of continuous probability distributions defined on the interval [0, 1] or (0, 1) in terms of two positive parameters, denoted by alpha (α) and beta (β), that appear as exponents of the variable and its complement to 1, respectively, and control the shape of the distribution.

The beta distribution has been applied to model the behavior of random variables limited to intervals of finite length in a wide variety of disciplines. The beta distribution is a suitable model for the random behavior of percentages and proportions.

In Bayesian inference, the beta distribution is the conjugate prior probability distribution for the Bernoulli, binomial, negative binomial, and geometric distributions.

The formulation of the beta distribution discussed here is also known as the beta distribution of the first kind, whereas beta distribution of the second kind is an alternative name for the beta prime distribution. The generalization to multiple variables is called a Dirichlet distribution.

The probability density function (PDF) of the beta distribution, for ${\displaystyle 0\leq x\leq 1}$ or ${\displaystyle 0<x<1}$, and shape parameters ${\displaystyle \alpha }$, ${\displaystyle \beta >0}$, is a power function of the variable ${\displaystyle x}$ and of its reflection ${\displaystyle (1-x)}$ as follows:

$${\displaystyle {\begin{aligned}f(x;\alpha ,\beta )&=\mathrm {constant} \cdot x^{\alpha -1}(1-x)^{\beta -1}\\[3pt]&={\frac {x^{\alpha -1}(1-x)^{\beta -1}}{\displaystyle \int _{0}^{1}u^{\alpha -1}(1-u)^{\beta -1}\,du}}\\[6pt]&={\frac {\Gamma (\alpha +\beta )}{\Gamma (\alpha )\Gamma (\beta )}}\,x^{\alpha -1}(1-x)^{\beta -1}\\[6pt]&={\frac {1}{\mathrm {B} (\alpha ,\beta )}}x^{\alpha -1}(1-x)^{\beta -1}\end{aligned}}}$$

where ${\displaystyle \Gamma (z)}$ is the gamma function. The beta function, ${\displaystyle \mathrm {B} }$, is a normalization constant to ensure that the total probability is 1. In the above equations ${\displaystyle x}$ is a realization—an observed value that actually occurred—of a random variable ${\displaystyle X}$.

Several authors, including N. L. Johnson and S. Kotz, use the symbols ${\displaystyle p}$ and ${\displaystyle q}$ (instead of ${\displaystyle \alpha }$ and ${\displaystyle \beta }$) for the shape parameters of the beta distribution, reminiscent of the symbols traditionally used for the parameters of the Bernoulli distribution, because the beta distribution approaches the Bernoulli distribution in the limit when both shape parameters ${\displaystyle \alpha }$ and ${\displaystyle \beta }$ approach zero.