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Gamma Distribution

NIST/SEMATECH Section 1.3.6.6.11 Gamma Distribution

PDF

Gamma(2, 1) PDF 0 1 2 3 4 5 6 7 8 9 x 0 0.1 0.2 0.3 0.4 f(x)

CDF

Gamma(2, 1) CDF 0 1 2 3 4 5 6 7 8 9 x 0 0.2 0.4 0.6 0.8 1 F(x)

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PDF

CDF

Formulas

Probability Density Function

f(x)=xα−1 e−x/ββα Γ(α),x>0f(x) = \frac{x^{\alpha-1}\,e^{-x/\beta}}{\beta^\alpha\,\Gamma(\alpha)}, \quad x > 0

Cumulative Distribution Function

F(x)=γ(α, x/β)Γ(α)F(x) = \frac{\gamma(\alpha,\, x/\beta)}{\Gamma(\alpha)}

Properties

Mean

αβ\alpha\beta

Variance

αβ2\alpha\beta^2

Overview

The gamma distribution is a two-parameter family of continuous distributions with shape α\alpha and scale β\beta that generalizes the exponential and chi-square distributions. It is used to model waiting times, rainfall amounts, and insurance claims.

Related Distributions

Chi-Square Distribution Exponential Distribution Beta Distribution