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

$$F\equiv {X/m\over Y/n}$$ (1)

is distributed as Snedecor's $F$-distribution with $m$ and $n$ degrees of freedom

$$f(F(m,n)) = {\Gamma\left({{\textstyle{m+n\over 2}}}\right)\l... ...}}}\right)\left({1+{\textstyle{m\over n}} F}\right)^{(m+n)/2}}$$ (2)

for $0<F<\infty$. The Moments about 0 are

$$\mu_1' = {n\over n-2}$$ (3)

$$\mu_2' = {n^2(m+2)\over m(n-2)(n-4)}$$ (4)

$$\mu_3' = {n^3(m+2)(m+4)\over m^2(n-2)(n-4)(n-6)}$$ (5)

$$\mu_4' = {n^4(m+2)(m+4)(m+6)\over m^3(n-2)(n-4)(n-6)(n-8)},$$ (6)

so the Moments about the Mean are given by

$$\mu_2 = {2n^2(m+n-2)\over m(n-2)^2(n-4)}$$ (7)

$$\mu_3 = {8n^3(m+n-2)(2m+n-2)\over m^2(n-2)^3(n-4)(n-6)}$$ (8)

$$\mu_4 = {12n^4(m+n-2)\over m^3(n-2)^4(n-4)(n-6)(n-8)} g(m,n),$$ (9)

where

$$g(m,n)=mn^2+4n^2+m^2n+8mn-16n+10m^2-20m+16,$$ (10)

and the Mean, Variance, Skewness, and Kurtosis are

$$\mu = \mu_1'={n\over n-2}$$ (11)

$$\sigma^2 = {2n^2(m+n-2)\over m(n-2)^2(n-4)}$$ (12)

$$\gamma_1 = {\mu_3\over\sigma^3} = 2\sqrt{2(n-4)\over m(m+n-2)} {2m+n-2\over n-6}$$ (13)

$$\gamma_2 = {\mu_4\over\sigma^4}-3$$

$$= {12h(m,n)\over m(m+n-2)(n-6)(n-8)},$$ (14)

where

$$h(m,n)\equiv n^3+5mn^2-8n^2+5m^2n-32mn+20n-22m^2+44m-16.$$ (15)

Letting

$$w\equiv {{mF\over n}\over 1+{mF\over n}}$$ (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.