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A two-parameter family of univariate distributions defined on the interval $[0,1]$.

The "Beta distribution" is a two-parameter family of continuous univariate distributions defined on the interval $[0,1]$. The probability density function of the distribution is

$$f_X(x;\alpha,\beta) = \frac{\Gamma(\alpha + \beta) }{\Gamma(\alpha) \Gamma(\beta) } x^{\alpha -1}(1-x)^{\beta-1}$$

with positive parameters $\alpha$ and $\beta$.

A common use of the distribution is in Bayesian statistics as a prior for the Binomial distribution. The Beta distribution is also used in beta regression, which can be useful when the dependent variable has a floor or ceiling effect or is bounded.

The distribution can be extended to represent random variables with support other than $[0,1]$, by using its four-parameter variant, that has the density function ($m$ = lower bound, $M$ = upper bound of the support)

$$f_X(x; \alpha, \beta, m, M) =\frac{\Gamma(\alpha + \beta) }{\Gamma(\alpha) \Gamma(\beta) } \frac{ (x-m)^{\alpha-1} (M-x)^{\beta-1} }{(M-m)^{\alpha+\beta-1}}\;$$

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