In probability theory and statistics, the beta distribution is a continuous probability distribution with the probability density function defined on the interval [0, 1]:
![f(x) = [\mbox{constant}]\cdot x^{\alpha-1}(1-x)^{\beta-1}.](a/../upload/math/0f41a5f791ac0041dd66f52be04a4403.png)
where α and β are parameters that must be greater than zero.
When the "constant" is included explicitly, the density looks like this:
where Γ and B are, respectively, the gamma function and the beta function.
The special case of the beta distribution when α = 1 and β = 1 is the standard uniform distribution.
The expected value and variance of a beta random variable X with parameters α and β are given by the formulae:
The kurtosis excess is:
On the other hand, with the expected value and variance of a beta random variable X given, the parameters α and β are calculated by the formulae:
![\alpha = \operatorname{E}(X) \left( \frac{\operatorname{E}(X)}{\operatorname{var}(X)} \left[ 1 - \operatorname{E}(X) \right] - 1 \right),](a/../upload/math/cb1cf26648a2c99997fe6ee452c34afc.png)
where 
and 
.
Cumulative distribution function
The cumulative distribution function is
where Bx(α,β) is the incomplete beta function and Ix(α,β) is the regularized incomplete beta function.
Shapes
The beta function can take on different shapes depending on the values of the two parameters:
- α = β = 1 is the uniform distribution
- α = β is symmetric about 1/2 (red & purple plots)
-
is U-shaped (red plot)
-
is strictly increasing (green plot)
-
is strictly decreasing (blue plot)
-
is unimodal (purple & black plots)
