Gammajakaumat

Gammajakaumat, known in English as gamma distributions, are a family of continuous probability distributions defined on the positive real numbers. They are described by two shape parameters and one scale or rate parameter and are used to model positive-valued data, waiting times, and various reliability processes. The term gammajakaumat is commonly used in Finnish statistical literature.

The distribution can be parameterized in two common ways. With shape α > 0 and scale θ > 0, its probability density function is f(x; α, θ) = x^(α-1) e^(-x/θ) / (Γ(α) θ^α) for x > 0. Equivalently, with shape α > 0 and rate β > 0, the density is f(x; α, β) = β^α x^(α-1) e^(-β x) / Γ(α) for x > 0, where β = 1/θ. The two parameterizations are related by θ = 1/β.

Moments and generating functions follow standard formulas. For shape α and scale θ, the mean is E[X] = α θ and the variance is Var[X] = α θ^2. The moment generating function is M(t) = (1 − θ t)^(-α) for t < 1/θ. In rate form, M(t) = (β/(β − t))^α for t < β.

Gamma distributions relate to other common distributions. Exponential distributions are gamma distributions with α = 1. The gamma with integer α is called Erlang. The gamma distribution also describes the waiting time until the α-th event in a Poisson process, and a chi-square distribution with ν degrees of freedom is a gamma distribution with α = ν/2 and θ = 2.

Parameter estimation commonly uses method of moments or maximum likelihood. Method of moments yields α̂ = m^2 / s^2 and θ̂ = s^2 / m, where m is the sample mean and s^2 the sample variance (rate form uses α̂ = μ^2 / σ^2 and β̂ = μ / σ^2). Gammajakaumat are widely used in reliability, queuing theory, Bayesian inference, and modeling of positive data.