Non-Central Chi-Square Distribution
X ~ χ²_k(λ)
Noncentral / Sum-of-Squared-Normals Family • Published July 22, 2026
The exact non-null distribution behind every chi-square test's power calculation: history from Fisher's 1920s treatment of the goodness-of-fit test's power, a Poisson-mixture density over the already-published Chi-Square's own densities with an exact λ=0 tie confirmed numerically, mean and variance formulas verified to ten decimal places, no closed-form maximum-likelihood estimator for either parameter, and simulation via the sum-of-squared-shifted-Normals construction that defines the distribution itself.
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About This Reference Sheet
The exact non-null distribution behind every chi-square test's power calculation: history from Fisher's 1920s treatment of the goodness-of-fit test's power, a Poisson-mixture density over the already-published Chi-Square's own densities with an exact λ=0 tie confirmed numerically, mean and variance formulas verified to ten decimal places, no closed-form maximum-likelihood estimator for either parameter, and simulation via the sum-of-squared-shifted-Normals construction that defines the distribution itself.
Support
x ∈ [0, ∞)
Parameters
k > 0 (degrees of freedom), λ ≥ 0 (noncentrality)