Editing F-distribution (section)

==Related distributions==
===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.

===In general===
*If <math>X \sim \chi^2_{d_1}</math> and <math>Y \sim \chi^2_{d_2}</math> ([[Chi squared distribution]]) are [[independence (probability theory)|independent]], then <math> \frac{X / d_1}{Y / d_2} \sim \mathrm{F}(d_1, d_2)</math>
*If <math>X_k \sim \Gamma(\alpha_k,\beta_k)\,</math> ([[Gamma distribution]]) are independent, then <math> \frac{\alpha_2\beta_1 X_1}{\alpha_1\beta_2 X_2} \sim \mathrm{F}(2\alpha_1, 2\alpha_2)</math>
*If <math>X \sim \operatorname{Beta}(d_1/2,d_2/2)</math> ([[Beta distribution]]) then <math>\frac{d_2 X}{d_1(1-X)} \sim \operatorname{F}(d_1,d_2)</math>
*Equivalently, if <math>X \sim F(d_1, d_2)</math>, then <math>\frac{d_1 X/d_2}{1+d_1 X/d_2} \sim \operatorname{Beta}(d_1/2,d_2/2)</math>.
*If <math>X \sim F(d_1, d_2)</math>, then <math>\frac{d_1}{d_2}X</math> has a [[beta prime distribution]]: <math>\frac{d_1}{d_2}X \sim \operatorname{\beta^\prime}\left(\tfrac{d_1}{2},\tfrac{d_2}{2}\right)</math>.
*If <math>X \sim F(d_1, d_2)</math> then <math>Y = \lim_{d_2 \to \infty} d_1 X</math> has the [[chi-squared distribution]] <math>\chi^2_{d_1}</math>
*<math>F(d_1, d_2)</math> is equivalent to the scaled [[Hotelling's T-squared distribution]] <math>\frac{d_2}{d_1(d_1+d_2-1)} \operatorname{T}^2 (d_1, d_1 +d_2-1) </math>.
*If <math>X \sim F(d_1, d_2)</math> then <math>X^{-1} \sim F(d_2, d_1)</math>.
*If <math>X\sim t_{(n)}</math> — [[Student's t-distribution]] — then: <math display="block">\begin{align}
X^{2} &\sim \operatorname{F}(1, n) \\
X^{-2} &\sim \operatorname{F}(n, 1)
\end{align}</math>
*''F''-distribution is a special case of type 6 [[Pearson distribution]]
*If <math>X</math> and <math>Y</math> are independent, with <math>X,Y\sim</math> [[Laplace distribution|Laplace(''μ'', ''b'')]] then <math display="block"> \frac{|X-\mu|}{|Y-\mu|} \sim \operatorname{F}(2,2) </math>
*If <math>X\sim F(n,m)</math> then <math>\tfrac{\log{X}}{2} \sim \operatorname{FisherZ}(n,m)</math> ([[Fisher's z-distribution]])
*The [[noncentral F-distribution|noncentral ''F''-distribution]] simplifies to the ''F''-distribution if <math>\lambda=0</math>.
*The doubly [[noncentral F-distribution|noncentral ''F''-distribution]] simplifies to the ''F''-distribution if <math> \lambda_1 = \lambda_2 = 0 </math>
*If <math>\operatorname{Q}_X(p)</math> is the quantile ''p'' for <math>X\sim F(d_1,d_2)</math> and <math>\operatorname{Q}_Y(1-p)</math> is the quantile <math>1-p</math> for <math>Y\sim F(d_2,d_1)</math>, then <math display="block">\operatorname{Q}_X(p)=\frac{1}{\operatorname{Q}_Y(1-p)}.</math>
* ''F''-distribution is an instance of [[ratio distributions]]
* [[Kendall's W|W]]-distribution<ref>{{Cite journal |last1=Mahmoudi |first1=Amin |last2=Javed |first2=Saad Ahmed |date=October 2022 |title=Probabilistic Approach to Multi-Stage Supplier Evaluation: Confidence Level Measurement in Ordinal Priority Approach |journal=Group Decision and Negotiation |language=en |volume=31 |issue=5 |pages=1051–1096 |doi=10.1007/s10726-022-09790-1 |issn=0926-2644 |pmc=9409630 |pmid=36042813}}</ref> is a unique parametrization of F-distribution.