Editing Normal distribution (section)

==== Operations on multiple independent normal variables ====
* Any [[linear combination]] of independent normal deviates is a normal deviate.
* If <math display=inline>X_1, X_2, \ldots, X_n</math> are independent standard normal random variables, then the sum of their squares has the [[chi-squared distribution]] with {{tmath|n}} degrees of freedom <math display=block>X_1^2 + \cdots + X_n^2 \sim \chi_n^2.</math>
* If <math display=inline>X_1, X_2, \ldots, X_n</math> are independent normally distributed random variables with means {{tmath|\mu}} and variances <math display=inline>\sigma^2</math>, then their [[sample mean]] is independent from the sample [[standard deviation]],<ref>{{cite journal|title=A Characterization of the Normal Distribution |last=Lukacs |first=Eugene |journal=[[The Annals of Mathematical Statistics]] |issn=0003-4851 |volume=13|issue=1 |year=1942 |pages=91–3 |jstor=2236166 |doi=10.1214/aoms/1177731647 |doi-access=free}}</ref> which can be demonstrated using [[Basu's theorem]] or [[Cochran's theorem]].<ref>{{cite journal |title=On Some Characterizations of the Normal Distribution | last1=Basu|first1=D. |last2=Laha|first2=R. G.|journal=[[Sankhyā (journal)|Sankhyā]]|issn=0036-4452| volume=13|issue=4|year=1954|pages=359–62| jstor=25048183}}</ref> The ratio of these two quantities will have the [[Student's t-distribution]] with <math display=inline>n-1</math> degrees of freedom: <math display=block>t = \frac{\overline X - \mu}{S/\sqrt{n}} = \frac{\frac{1}{n}(X_1+\cdots+X_n) - \mu}{\sqrt{\frac{1}{n(n-1)}\left[(X_1-\overline X)^2 + \cdots+(X_n-\overline X)^2\right]}} \sim t_{n-1}.</math>
* If <math display=inline>X_1, X_2, \ldots, X_n</math>, <math display=inline>Y_1, Y_2, \ldots, Y_m</math> are independent standard normal random variables, then the ratio of their normalized sums of squares will have the [[F-distribution]] with {{math|(''n'', ''m'')}} degrees of freedom:<ref>{{cite book |title=Testing Statistical Hypotheses |edition=2nd | first=E. L. | last=Lehmann | publisher=Springer |year=1997 | isbn=978-0-387-94919-2| page=199}}</ref> <math display=block>F = \frac{\left(X_1^2+X_2^2+\cdots+X_n^2\right)/n}{\left(Y_1^2+Y_2^2+\cdots+Y_m^2\right)/m} \sim F_{n,m}.</math>