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In probability theory and statistics, the beta distribution is a continuous probability distribution with the probability density function defined on the interval [0, 1]:
- <math> f(x) = [\mbox{constant}]\cdot x^{a-1}(1-x)^{b-1}.<math>
where a and b are parameters that must be greater than zero.
When the "constant" is included explicitly, the density looks like this:
- <math> f(x) = \frac{x^{a-1}(1-x)^{b-1}}{\int_0^1 u^{a-1} (1-u)^{b-1}\, du}
= \frac{\Gamma(a+b)}{\Gamma(a)\Gamma(b)}\, x^{a-1}(1-x)^{b-1}
= \frac{1}{B(a,b)}\, x^{a-1}(1-x)^{b-1} <math>
where Γ and B are respectively the gamma function and the beta function.
The special case of the beta distribution, when a = 1 and b = 1, is the standard uniform distribution.
The expected value and variance of a beta random variable X with parameters a and b are given by the formulae:
- <math> \mbox{E}(X) = \frac{a}{a+b}, <math>
- <math> \mbox{var}(X) = \frac{ab}{(a+b)^2(a+b+1)}. <math>
On the other hand, with the expected value and variance of a beta random variable X given, the parameters a and b are calculated by the formulae:
- <math> a = \mbox{E}(X)\left(\frac{\mbox{E}(X)}{\mbox{var}(X)}[1-\mbox{E}(X)]-1\right), <math>
- <math> b = a\frac{1-\mbox{E}(X)}{\mbox{E}(X)} <math>
where 0 < E(X) < 1 and 0 < var(X) < E(X) (1 − E(X)).
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