Betajakaumat

Betajakaumat, Finnish for beta distributions, are a family of continuous probability distributions supported on the interval [0, 1]. They are defined by two positive shape parameters, α and β, that control the distribution’s concentration near 0 and 1.

The probability density function is f(x; α, β) = x^{α−1} (1−x)^{β−1} / B(α,β) for 0 ≤ x ≤ 1, where B(α,β) is the Beta function, B(α,β) = Γ(α) Γ(β) / Γ(α+β).

Moments and shape: The mean is E[X] = α/(α+β) and the variance is Var[X] = αβ / [(α+β)^2 (α+β+1)]. When α>1 and β>1 the mode is (α−1)/(α+β−2). A common special case is α = β = 1, which yields the Uniform(0,1) distribution.

Representations and connections: If A ~ Gamma(α,1) and B ~ Gamma(β,1) are independent, then X = A/(A+B) follows Beta(α,β). The Beta distribution also appears as the conjugate prior for the probability parameter p of a Bernoulli or Binomial model; observing s successes and f failures updates α to α+s and β to β+f. In higher dimensions, the Dirichlet distribution generalizes this concept, describing a distribution over probability vectors.

Sampling and applications: Betajakaumat are used to model uncertainty about a probability in Bayesian inference and reliability analysis. They lead to the Beta-Binomial distribution when combined with a Binomial likelihood. Common sampling methods generate two Gamma variates and normalize, or use standard routines for the beta distribution.

Name origin: The name derives from the Beta function, central to its definition and properties.