Chi-squared distribution

Probability distribution and special case of gamma distribution

In probability theory and statistics, the χ 2 {\displaystyle \chi ^{2}} -distribution with k {\displaystyle k} degrees of freedom is the distribution of a sum of the squares of k {\displaystyle k} independent standard normal random variables. The chi-squared distribution χ k 2 {\displaystyle \chi _{k}^{2}} is a special case of the gamma distribution and the univariate Wishart distribution. Specifically if X ∼ χ k 2 {\displaystyle X\sim \chi _{k}^{2}} then X ∼ Gamma ( α = k 2 , θ = 2) {\textstyle X\sim {\text{Gamma}}(\alpha ={\frac {k}{2}},\theta =2)} (where α {\displaystyle \alpha } is the shape parameter and θ {\displaystyle \theta } the scale parameter of the gamma distribution) and X ∼ W 1 ( 1 , k) {\displaystyle X\sim {\text{W}}_{1}(1,k)} .

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