Editing F-distribution (section)

===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.