Gamma and chi-squared distributions relation

Gamma distribution

A continuous variable X > 0 has a gamma distribution with shape α > 0 and scale β > 0 if

f(x) = x^(α−1) e^(−x/β) / (βᵅ Γ(α)), x > 0.

Mean = αβ, variance = αβ². It models waiting times and lifetimes.

Chi-squared distribution

A continuous variable χ² > 0 has a chi-squared distribution with ν degrees of freedom if

f(x) = x^(ν/2 − 1) e^(−x/2) / (2^(ν/2) Γ(ν/2)), x > 0.

Mean = ν, variance = 2ν. It arises as the sum of squares of ν independent standard normal variables: χ² = Z₁² + Z₂² + … + Zν². It is used in tests of variance, goodness of fit and independence.

Relationship

The chi-squared distribution is a special case of the gamma distribution obtained by putting

α = ν/2 and β = 2

Substituting these in the gamma pdf gives exactly the chi-squared pdf. Checking the moments: mean αβ = (ν/2)(2) = ν, and variance αβ² = (ν/2)(4) = 2ν.

Also, the exponential distribution is gamma with α = 1, so chi-squared with 2 degrees of freedom is exponential with mean 2.