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Snedecor's F-Distribution

If a random variable $X$ has a Chi-Squared Distribution with $m$ degrees of freedom (${\chi_m}^2$) and a random variable $Y$ has a Chi-Squared Distribution with $n$ degrees of freedom (${\chi_n}^2$), and $X$ and $Y$ are independent, then

\begin{displaymath}
F\equiv {X/m\over Y/n}
\end{displaymath} (1)

is distributed as Snedecor's $F$-distribution with $m$ and $n$ degrees of freedom
\begin{displaymath}
f(F(m,n)) = {\Gamma\left({{\textstyle{m+n\over 2}}}\right)\l...
...}}}\right)\left({1+{\textstyle{m\over n}} F}\right)^{(m+n)/2}}
\end{displaymath} (2)

for $0<F<\infty$. The Moments about 0 are
$\displaystyle \mu_1'$ $\textstyle =$ $\displaystyle {n\over n-2}$ (3)
$\displaystyle \mu_2'$ $\textstyle =$ $\displaystyle {n^2(m+2)\over m(n-2)(n-4)}$ (4)
$\displaystyle \mu_3'$ $\textstyle =$ $\displaystyle {n^3(m+2)(m+4)\over m^2(n-2)(n-4)(n-6)}$ (5)
$\displaystyle \mu_4'$ $\textstyle =$ $\displaystyle {n^4(m+2)(m+4)(m+6)\over m^3(n-2)(n-4)(n-6)(n-8)},$ (6)

so the Moments about the Mean are given by
$\displaystyle \mu_2$ $\textstyle =$ $\displaystyle {2n^2(m+n-2)\over m(n-2)^2(n-4)}$ (7)
$\displaystyle \mu_3$ $\textstyle =$ $\displaystyle {8n^3(m+n-2)(2m+n-2)\over m^2(n-2)^3(n-4)(n-6)}$ (8)
$\displaystyle \mu_4$ $\textstyle =$ $\displaystyle {12n^4(m+n-2)\over m^3(n-2)^4(n-4)(n-6)(n-8)} g(m,n),$ (9)

where


\begin{displaymath}
g(m,n)=mn^2+4n^2+m^2n+8mn-16n+10m^2-20m+16,
\end{displaymath} (10)

and the Mean, Variance, Skewness, and Kurtosis are
$\displaystyle \mu$ $\textstyle =$ $\displaystyle \mu_1'={n\over n-2}$ (11)
$\displaystyle \sigma^2$ $\textstyle =$ $\displaystyle {2n^2(m+n-2)\over m(n-2)^2(n-4)}$ (12)
$\displaystyle \gamma_1$ $\textstyle =$ $\displaystyle {\mu_3\over\sigma^3} = 2\sqrt{2(n-4)\over m(m+n-2)} {2m+n-2\over n-6}$ (13)
$\displaystyle \gamma_2$ $\textstyle =$ $\displaystyle {\mu_4\over\sigma^4}-3$  
  $\textstyle =$ $\displaystyle {12h(m,n)\over m(m+n-2)(n-6)(n-8)},$ (14)

where


\begin{displaymath}
h(m,n)\equiv n^3+5mn^2-8n^2+5m^2n-32mn+20n-22m^2+44m-16.
\end{displaymath} (15)

Letting
\begin{displaymath}
w\equiv {{mF\over n}\over 1+{mF\over n}}
\end{displaymath} (16)

gives a Beta Distribution.

See also Beta Distribution, Chi-Squared Distribution, Student's t-Distribution


References

Beyer, W. H. CRC Standard Mathematical Tables, 28th ed. Boca Raton, FL: CRC Press, p. 536, 1987.



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© 1996-9 Eric W. Weisstein
1999-05-26