Beta distribution - Wikipedia

Abstraction: Continuous distribution on [0,1] conjugate prior for Bernoulli/binomial

Key points:

• Defined on [0,1] by two shape parameters alpha and beta; PDF is proportional to x^(alpha-1)(1-x)^(beta-1) normalized by the beta function
• Conjugate prior for Bernoulli, binomial, negative binomial, and geometric distributions in Bayesian inference
• Mean = alpha/(alpha+beta); variance = alphabeta / ((alpha+beta)^2 (alpha+beta+1)); symmetric when alpha=beta
• Can be reparametrized by mean mu and sample size nu = alpha+beta for intuitive Bayesian updating
• Limiting cases: approaches Bernoulli as both parameters approach zero; becomes degenerate (point mass at 0 or 1) when ratio alpha/beta goes to 0 or infinity
• Minimum possible excess kurtosis (-2) achieved at alpha=beta=0, corresponding to maximum variance of 1/4
• The generalization to multiple variables is the Dirichlet distribution

Connections: Probability Distributions · Bayesian Inference · Conjugate Prior

Source: http://en.wikipedia.org/wiki/Beta_distribution