F Distribution
X ~ F_(d1, d2)
Ratio-of-Chi-Squares Family • Published July 20, 2026
The variance-ratio distribution behind every ANOVA and regression F-test: history from Fisher's 1924 variance-ratio work through George Snedecor's 1934 naming, its exact link to the already-published Beta distribution via a genuine change of variable, why no moment generating function exists, a verified closed-form 2x2 Fisher information matrix in the trigamma function, a two-dimensional transcendental maximum-likelihood system with no closed form, and simulation via its own defining ratio-of-Chi-Squares construction.
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About This Reference Sheet
The variance-ratio distribution behind every ANOVA and regression F-test: history from Fisher's 1924 variance-ratio work through George Snedecor's 1934 naming, its exact link to the already-published Beta distribution via a genuine change of variable, why no moment generating function exists, a verified closed-form 2x2 Fisher information matrix in the trigamma function, a two-dimensional transcendental maximum-likelihood system with no closed form, and simulation via its own defining ratio-of-Chi-Squares construction.
Support
x ∈ [0, ∞)
Parameters
d1 > 0 (numerator df), d2 > 0 (denominator df)