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-tests.[2][3][4][5]
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Fisher–Snedecor
Probability density function |
Cumulative distribution function |
Parameters |
d1, d2 > 0 deg. of freedom |
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Support |
if , otherwise ![{\displaystyle x\in [0,+\infty )\;}](//wikimedia.org/api/rest_v1/media/math/render/svg/3ca3880f232f03079bb34e7839e668a75fac30b7) |
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PDF |
![{\displaystyle {\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)}}\!}](//wikimedia.org/api/rest_v1/media/math/render/svg/68bf5e8aeba175b3d855183a250e3843c715a416) |
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CDF |
![{\displaystyle I_{\frac {d_{1}x}{d_{1}x+d_{2}}}\left({\tfrac {d_{1}}{2}},{\tfrac {d_{2}}{2}}\right)}](//wikimedia.org/api/rest_v1/media/math/render/svg/7d62d2ffb6f95571e911c905fd6e04e94eb18377) |
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Mean |
![{\displaystyle {\frac {d_{2}}{d_{2}-2}}\!}](//wikimedia.org/api/rest_v1/media/math/render/svg/42bc770a649bafe0249c3f8c4614b45f42241f0a) for d2 > 2 |
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Mode |
![{\displaystyle {\frac {d_{1}-2}{d_{1}}}\;{\frac {d_{2}}{d_{2}+2}}}](//wikimedia.org/api/rest_v1/media/math/render/svg/fa498a3d7ae8205dd2c0dc4a1a325dbec007aa83) for d1 > 2 |
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Variance |
![{\displaystyle {\frac {2\,d_{2}^{2}\,(d_{1}+d_{2}-2)}{d_{1}(d_{2}-2)^{2}(d_{2}-4)}}\!}](//wikimedia.org/api/rest_v1/media/math/render/svg/ce64d1edab4849983bd9d1590b30ce8a3d65ca73) for d2 > 4 |
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Skewness |
![{\displaystyle {\frac {(2d_{1}+d_{2}-2){\sqrt {8(d_{2}-4)}}}{(d_{2}-6){\sqrt {d_{1}(d_{1}+d_{2}-2)}}}}\!}](//wikimedia.org/api/rest_v1/media/math/render/svg/ac47c2f77fbcda51696e9f0819ff405c7f4c5b47) for d2 > 6 |
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Excess kurtosis |
see text |
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Entropy |
![{\displaystyle \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)+\!}](//wikimedia.org/api/rest_v1/media/math/render/svg/a507d266e4e8fc6eeecff472ac0cd3c55b8ed681)
![{\displaystyle \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)\!}](//wikimedia.org/api/rest_v1/media/math/render/svg/f87cb3fbf968cbc40c36c51e6cc777bdfcfacb93)
[1] |
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MGF |
does not exist, raw moments defined in text and in [2][3] |
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CF |
see text |
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