Beta Distribution
- is a family of continuous probability distributions defined on the interval [0, 1] parametrized by two positive shape parameters, 𝛼 and 𝛽, that control the shape of the distribution.
- the generalization to multiple variables is called a Dirichlet Distribution
Probability Density Function
- 𝑓(𝑥) = (1 - 𝑥)𝛽-1𝑥𝛼-1/ 𝐵(𝛼,𝛽)
where:
- 𝐵(𝛼,𝛽) is the beta function, i.e.:
- 𝐵(𝛼,𝛽) = [𝛤(𝛼)𝛤(𝛽)] / 𝛤(𝛼+𝛽)
- 𝐵(𝛼,𝛽) = [(𝛼-1)!(𝛽-1)!] / (𝛼+𝛽-1)!
where:
- 𝛤 is the gamma function, i.e. 𝛤(𝑛) = (𝑛 - 1)!
Probability Density Function
/beta-distribution/beta-distribution-probability-density-function.png)
Cumulative Distribution Function
/beta-distribution/beta-distribution-cumulative-distribution-function.png)