Let 
be a sum of squares of 
independent normal standardized variates 
, and suppose 
where 
is a quadratic form in the 
, distributed as chi-squared
with 
degrees of freedom. Then 
is distributed as 
with 
degrees of freedom and
is independent of 
.
The converse of this theorem is known as Cochran's
theorem.
Fisher's Theorem
See also
Chi-Squared Distribution, Cochran's TheoremExplore with Wolfram|Alpha
Cite this as:
Weisstein, Eric W. "Fisher's Theorem." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/FishersTheorem.html