Editing F-distribution (section)

===Relation to the chi-squared distribution===
In instances where the ''F''-distribution is used, for example in the [[analysis of variance]], independence of <math>U_1</math> and <math>U_2</math> (defined above) might be demonstrated by applying [[Cochran's theorem]].

Equivalently, since the [[chi-squared distribution]] is the sum of squares of [[Independence (probability theory)|independent]] [[standard normal]] random variables, the random variable of the ''F''-distribution may also be written

<math display="block">X = \frac{s_1^2}{\sigma_1^2} \div \frac{s_2^2}{\sigma_2^2},</math>

where <math>s_1^2 = \frac{S_1^2}{d_1}</math> and <math>s_2^2 = \frac{S_2^2}{d_2}</math>, <math>S_1^2</math> is the sum of squares of <math>d_1</math> random variables from normal distribution <math>N(0,\sigma_1^2)</math> and <math>S_2^2</math> is the sum of squares of <math>d_2</math> random variables from normal distribution <math>N(0,\sigma_2^2)</math>.

In a [[frequentist]] context, a scaled ''F''-distribution therefore gives the probability <math>p(s_1^2/s_2^2 \mid \sigma_1^2, \sigma_2^2)</math>, with the ''F''-distribution itself, without any scaling, applying where <math>\sigma_1^2</math>  is being taken equal to <math>\sigma_2^2</math>. This is the context in which the ''F''-distribution most generally appears in [[F-test|''F''-tests]]: where the null hypothesis is that two independent normal variances are equal, and the observed sums of some appropriately selected squares are then examined to see whether their ratio is significantly incompatible with this null hypothesis.

The quantity <math>X</math> has the same distribution in Bayesian statistics, if an uninformative rescaling-invariant [[Jeffreys prior]] is taken for the [[prior probability|prior probabilities]] of <math>\sigma_1^2</math> and <math>\sigma_2^2</math>.<ref>{{cite book |first=G. E. P. |last=Box |first2=G. C. |last2=Tiao |year=1973 |title=Bayesian Inference in Statistical Analysis |publisher=Addison-Wesley |page=110 |isbn=0-201-00622-7 }}</ref> In this context, a scaled ''F''-distribution thus gives the posterior probability <math>p(\sigma^2_2 /\sigma_1^2 \mid s^2_1, s^2_2)</math>, where the observed sums <math>s^2_1</math> and <math>s^2_2</math> are now taken as known.