Beta distribution
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Template:Probability distribution In probability theory and statistics, the beta distribution is a continuous probability distribution with the probability density function (pdf) defined on the interval [0, 1]:
- <math> f(x) = \frac{1}{\mathrm{B}(\alpha,\beta)} x^{\alpha-1}(1-x)^{\beta-1}<math>
where <math>\alpha<math> and <math>\beta<math> are parameters that must be greater than zero and <math>\mathrm{B}<math> is the beta function.
The beta function is a normalization constant to ensure that the integral of the pdf is unity:
- <math> f(x) = \frac{x^{\alpha-1}(1-x)^{\beta-1}}{\int_0^1 u^{\alpha-1} (1-u)^{\beta-1}\, du} \!<math>
- <math>= \frac{\Gamma(\alpha+\beta)}{\Gamma(\alpha)\Gamma(\beta)}\, x^{\alpha-1}(1-x)^{\beta-1}\!<math>
- <math>= \frac{1}{\mathrm{B}(\alpha,\beta)}\, x^{\alpha-1}(1-x)^{\beta-1}\!<math>
where <math>\Gamma<math> is the gamma function.
The special case of the beta distribution when <math>\alpha=1<math> and <math>\beta=1<math> is the standard uniform distribution.
The expected value and variance of a beta random variable <math>X<math> with parameters <math>\alpha<math> and <math>\beta<math> are given by the formulae:
- <math> \operatorname{E}(X) = \frac{\alpha}{\alpha+\beta}, <math>
- <math> \operatorname{var}(X) = \frac{\alpha \beta}{(\alpha+\beta)^2(\alpha+\beta+1)}.<math>
The kurtosis excess is:
- <math>6\,\frac{\alpha^3-\alpha^2(2\beta-1)+\beta^2(\beta+1)-2\alpha\beta(\beta+2)}
{\alpha \beta (\alpha+\beta+2) (\alpha+\beta+3)}\!<math>
On the other hand, with the expected value and variance of a beta random variable <math>X<math> given, the parameters <math>\alpha<math> and <math>\beta<math> are calculated by the formulae:
- <math>
\alpha = \operatorname{E}(X) \left(
\frac{\operatorname{E}(X)}{\operatorname{var}(X)}
\left[
1 - \operatorname{E}(X)
\right]
- 1
\right),<math>
- <math>\beta = \alpha \frac{1-\operatorname{E}(X)}{\operatorname{E}(X)}<math>
where <math>0 < \operatorname{E}(X) < 1<math> and <math>0 < \operatorname{var}(X) < \operatorname{E}(X) (1 - \operatorname{E}(X))<math>.
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Cumulative distribution function
The cumulative distribution function is
- <math>F(x) = \frac{\mathrm{B}_x(\alpha,\beta)}{\mathrm{B}(\alpha,\beta)} = I_x(\alpha,\beta) \!<math>
where <math>\mathrm{B}_x(\alpha,\beta)<math> is the incomplete beta function and <math>I_x(\alpha,\beta)<math> is the regularized incomplete beta function.
Shapes
The beta function can take on different shapes depending on the values of the two parameters:
- <math>\alpha = \beta = 1<math> is the uniform distribution
- <math>\alpha = \beta<math> is symmetric about 1/2 (red & purple plots)
- <math>\alpha < 1,\ \beta > 1<math> is U-shaped (red plot)
- <math>\alpha > 1,\ \beta = 1<math> is strictly increasing (green plot)
- <math>\alpha = 1,\ \beta > 1<math> is strictly decreasing (blue plot)
- <math>\alpha > 1,\ \beta > 1<math> is unimodal (purple & black plots)
Related distributions
- <math>X \sim \mathrm{Uniform}(0,1)<math> is a uniform distribution if <math>X \sim \mathrm{Beta}(\alpha = 1, \beta = 1)<math>.de:Betaverteilung
es:Distribución beta it:Variabile casuale Beta pl:Rozkład beta
Applications
Beta distributions are used extensively in Bayesian statistics, since the beta distribution is the conjugate prior distribution to the binomial distribution.