Table of Contents
What Is the Beta Distribution?
The Beta distribution is a continuous probability distribution defined on [0,1], parameterized by two shape parameters alpha and beta. It is incredibly versatile and can model a wide variety of shapes including uniform, U-shaped, J-shaped, and bell-shaped distributions.
Common applications include Bayesian statistics (as a conjugate prior for binomial proportions), modeling probabilities, project management (PERT estimation), and A/B testing.
Formulas
Shape Parameters
| α | β | Shape |
|---|---|---|
| 1 | 1 | Uniform |
| 2 | 5 | Right-skewed |
| 5 | 2 | Left-skewed |
| 5 | 5 | Bell-shaped |
| 0.5 | 0.5 | U-shaped |
FAQ
Why is Beta used in Bayesian statistics?
It is the conjugate prior for binomial likelihood. If your prior is Beta(α,β) and you observe k successes in n trials, the posterior is Beta(α+k, β+n-k). This makes updating beliefs computationally elegant.
What does Beta(1,1) represent?
Beta(1,1) is the uniform distribution on [0,1], representing complete ignorance about a probability parameter.