Chi-Square Distribution
X ~ χ²_k
Sum-of-Squared-Normals Family • Published July 20, 2026
The sum-of-squared-Normals distribution behind Pearson's 1900 goodness-of-fit test: history from Helmert's 1875-76 geodetic-error work, its exact identity as the already-published Gamma density with shape k/2 and rate 1/2, the full property list including a closed-form Fisher information via the trigamma function, a transcendental digamma maximum-likelihood equation verified against direct numerical optimization, and simulation via its own defining sum-of-squared-Normals construction.
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About This Reference Sheet
The sum-of-squared-Normals distribution behind Pearson's 1900 goodness-of-fit test: history from Helmert's 1875-76 geodetic-error work, its exact identity as the already-published Gamma density with shape k/2 and rate 1/2, the full property list including a closed-form Fisher information via the trigamma function, a transcendental digamma maximum-likelihood equation verified against direct numerical optimization, and simulation via its own defining sum-of-squared-Normals construction.
Support
x ∈ [0, ∞)
Parameters
k > 0 (degrees of freedom)