standardgamma

Standard gamma, in statistics, refers to the standard gamma distribution: a gamma distribution with shape parameter α > 0 and scale θ = 1. It is a continuous probability distribution on the positive real numbers, used to model positive-valued skewed data. Its probability density function is f(x; α) = x^{α-1} e^{-x} / Γ(α) for x > 0, where Γ(α) is the gamma function.

Key properties: The mean is α and the variance is α. The moment generating function is M_X(t) = (1 − t)^{−α} for t < 1. When α = 1, the standard gamma reduces to the standard exponential distribution with mean 1. If X ~ Gamma(α, 1) and Y ~ Gamma(β, 1) are independent, then X + Y ~ Gamma(α + β, 1).

Relation to other distributions: 2X has a chi-square distribution with 2α degrees of freedom, i.e., 2X ~ χ²_{2α}. The gamma distribution serves as a flexible model in Bayesian statistics, reliability engineering, and queueing theory, and forms a family with various parameterizations and connections to other distributions.

Parameterizations: Some texts specify a rate parameter β > 0 instead of a scale; X ~ Gamma(α, β) with pdf f(x) = β^α x^{α-1} e^{−β x} / Γ(α). The standard gamma uses β = 1 (scale θ = 1). The gamma function Γ(z) generalizes the factorial, with Γ(n) = (n−1)! for integers and Γ(z) = ∫_0^∞ t^{z−1} e^{−t} dt for Re(z) > 0.

Summary: The standard gamma is a widely used baseline distribution for modeling positive data and as a building block in statistical models, defined by shape and unit scale, and related through the gamma function to broader mathematical and probabilistic frameworks.

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