Rozkład F Snedecora – rozkład prawdopodobieństwa zmiennej losowej F o d1, d2 stopniach swobody.
d 1 > 0 , d 2 > 0 {\displaystyle d_{1}>0,\ d_{2}>0} stopni swobody
x ∈ [ 0 ; + ∞ ) {\displaystyle x\in [0;+\infty )}
( d 1 x ) d 1 d 2 d 2 ( d 1 x + d 2 ) d 1 + d 2 x B ( d 1 2 , d 2 2 ) {\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)}}}
I d 1 x d 1 x + d 2 ( d 1 / 2 , d 2 / 2 ) {\displaystyle I_{\frac {d_{1}x}{d_{1}x+d_{2}}}(d_{1}/2,d_{2}/2)}
d 2 d 2 − 2 {\displaystyle {\frac {d_{2}}{d_{2}-2}}} dla d 2 > 2 {\displaystyle {\text{ dla }}d_{2}>2}
d 1 − 2 d 1 d 2 d 2 + 2 {\displaystyle {\frac {d_{1}-2}{d_{1}}}\;{\frac {d_{2}}{d_{2}+2}}} dla d 1 > 2 {\displaystyle {\text{ dla }}d_{1}>2}
2 d 2 2 ( d 1 + d 2 − 2 ) d 1 ( d 2 − 2 ) 2 ( d 2 − 4 ) {\displaystyle {\frac {2\,d_{2}^{2}\,(d_{1}+d_{2}-2)}{d_{1}(d_{2}-2)^{2}(d_{2}-4)}}} dla d 2 > 4 {\displaystyle {\text{ dla }}d_{2}>4}
( 2 d 1 + d 2 − 2 ) 8 ( d 2 − 4 ) ( d 2 − 6 ) d 1 ( d 1 + d 2 − 2 ) {\displaystyle {\frac {(2d_{1}+d_{2}-2){\sqrt {8(d_{2}-4)}}}{(d_{2}-6){\sqrt {d_{1}(d_{1}+d_{2}-2)}}}}} for d 2 > 6 {\displaystyle d_{2}>6}
12 ( 20 d 2 − 8 d 2 2 + d 2 3 + 44 d 1 − 32 d 1 d 2 d 1 ( d 2 − 6 ) ( d 2 − 8 ) ( d 1 + d 2 − 2 ) + {\displaystyle {\tfrac {12(20d_{2}-8d_{2}^{2}+d_{2}^{3}+44d_{1}-32d_{1}d_{2}}{d_{1}(d_{2}-6)(d_{2}-8)(d_{1}+d_{2}-2)}}+} + 5 d 2 2 d 1 − 22 d 1 2 + 5 d 2 d 1 2 − 16 ) d 1 ( d 2 − 6 ) ( d 2 − 8 ) ( d 1 + d 2 − 2 ) . {\displaystyle +{\tfrac {5d_{2}^{2}d_{1}-22d_{1}^{2}+5d_{2}d_{1}^{2}-16)}{d_{1}(d_{2}-6)(d_{2}-8)(d_{1}+d_{2}-2)}}.}
Ronald Fisher, George W. Snedecor
Jeżeli X i Y są niezależne oraz X ∼ χ 2 ( d 1 ) {\displaystyle X\sim \chi ^{2}(d_{1})\ {}} i Y ∼ χ 2 ( d 2 ) , {\displaystyle {}\ Y\sim \chi ^{2}(d_{2}),} to:
stopni swobody
