Editing F-distribution (section)

{{Short description|Continuous probability distribution}}
{{About|the central F-distribution|the generalized distribution|noncentral F-distribution|other uses|F-ratio (disambiguation){{!}}F-ratio}}
{{distinguish|text=[[F-statistics|''F''-statistics]] as used in population genetics}}
{{DISPLAYTITLE:''F''-distribution}}
{{Probability distribution
| name       = Fisher–Snedecor
| type       = density
| pdf_image  = [[Image:F-distribution pdf.svg|325px]]|
| cdf_image  = [[Image:F_dist_cdf.svg|325px]]|
| parameters = ''d''<sub>1</sub>, ''d''<sub>2</sub> > 0 deg. of freedom
| support    = <math>x \in (0, +\infty)\;</math> if <math>d_1 = 1</math>, otherwise <math>x \in [0, +\infty)\;</math>
| pdf        = <math>\frac{\sqrt{\frac{(d_1 x)^{d_1} d_2^{d_2}}{(d_1 x+d_2)^{d_1+d_2}}}}{x\,\mathrm{B}\!\left(\frac{d_1}{2},\frac{d_2}{2}\right)}\!</math>
| cdf        = <math>I_{\frac{d_1 x}{d_1 x + d_2}} \left(\tfrac{d_1}{2}, \tfrac{d_2}{2} \right)</math>
| mean       = <math>\frac{d_2}{d_2-2}\!</math><br /> for ''d''<sub>2</sub> > 2
| median     =
| mode       = <math>\frac{d_1-2}{d_1}\;\frac{d_2}{d_2+2}</math><br /> for ''d''<sub>1</sub> > 2
| variance   = <math>\frac{2\,d_2^2\,(d_1+d_2-2)}{d_1 (d_2-2)^2 (d_2-4)}\!</math><br /> for ''d''<sub>2</sub> > 4
| skewness   = <math>\frac{(2 d_1 + d_2 - 2) \sqrt{8 (d_2-4)}}{(d_2-6) \sqrt{d_1 (d_1 + d_2 -2)}}\!</math><br />for ''d''<sub>2</sub> > 6
| kurtosis   = ''see text''
| entropy    = <math>\begin{align}
& \ln \Gamma{\left(\tfrac{d_1}{2} \right)}
 + \ln \Gamma{\left(\tfrac{d_2}{2} \right)}
 - \ln \Gamma{\left(\tfrac{d_1+d_2}{2} \right)} \\
&+ \left(1-\tfrac{d_1}{2} \right) \psi{\left(1+\tfrac{d_1}{2} \right)}
 - \left(1+\tfrac{d_2}{2} \right) \psi{\left(1+\tfrac{d_2}{2} \right)} \\
&+ \left(\tfrac{d_1 + d_2}{2} \right) \psi{\left(\tfrac{d_1 + d_2}{2} \right)}
 + \ln \frac{d_2}{d_1}
\end{align}</math><ref name=lazo1978entropy>{{Cite journal|last1=Lazo |first1=A.V. |last2=Rathie |first2=P. |title=On the entropy of continuous probability distributions |journal=IEEE Transactions on Information Theory |volume=24 |number=1 |pages=120–122 |year=1978 |publisher=IEEE |doi=10.1109/tit.1978.1055832}}</ref>
| mgf        = ''does not exist, raw moments defined in text and in <ref name=johnson /><ref name=abramowitz />''
| char       = ''see text''
}}

In [[probability theory]] and [[statistics]], the '''''F''-distribution''' or '''''F''-ratio''', also known as '''Snedecor's ''F'' distribution''' or the '''Fisher–Snedecor distribution''' (after [[Ronald Fisher]] and [[George W. Snedecor]]), is a [[continuous probability distribution]] that arises frequently as the [[null distribution]] of a [[test statistic]], most notably in the [[analysis of variance]] (ANOVA) and other [[F-test|''F''-tests]].<ref name=johnson>{{cite book | last = Johnson | first = Norman Lloyd | author2 = Samuel Kotz | author3 = N. Balakrishnan | title = Continuous Univariate Distributions, Volume 2 (Section 27) | edition = 2nd | publisher = Wiley | year = 1995 | isbn = 0-471-58494-0}}</ref><ref name=abramowitz>{{Abramowitz_Stegun_ref|26|946}}</ref><ref>NIST (2006). [http://www.itl.nist.gov/div898/handbook/eda/section3/eda3665.htm Engineering Statistics Handbook – F Distribution]</ref><ref>{{cite book | last = Mood | first = Alexander | author2 = Franklin A. Graybill | author3 = Duane C. Boes | title = Introduction to the Theory of Statistics | edition = Third | pages = 246–249 | publisher = McGraw-Hill | year = 1974 | isbn = 0-07-042864-6}}</ref>