TOPICS
Search

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=(X/m)/(Y/n)
(1)

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

 f(F(m,n))=(Gamma((m+n)/2)(m/n)^(m/2)F^((m-2)/2))/(Gamma(m/2)Gamma(n/2)(1+m/nF)^((m+n)/2))
(2)

for 0<F<infty. The raw moments are

mu_1^'=n/(n-2)
(3)
mu_2^'=(n^2(m+2))/(m(n-2)(n-4))
(4)
mu_3^'=(n^3(m+2)(m+4))/(m^2(n-2)(n-4)(n-6))
(5)
mu_4^'=(n^4(m+2)(m+4)(m+6))/(m^3(n-2)(n-4)(n-6)(n-8)),
(6)

so the first few central moments are given by

mu_2=(2n^2(m+n-2))/(m(n-2)^2(n-4))
(7)
mu_3=(8n^3(m+n-2)(2m+n-2))/(m^2(n-2)^3(n-4)(n-6))
(8)
mu_4=(12n^4(m+n-2)[4(n-2)^2+m^2(n+10)+m(n-2)(n+10)])/(m^3(n-2)^4(n-4)(n-6)(n-8)),
(9)

and the mean, variance, skewness, and kurtosis excess are

mu=mu_1^'=n/(n-2)
(10)
sigma^2=(2n^2(m+n-2))/(m(n-2)^2(n-4))
(11)
gamma_1=2sqrt((2(n-4))/(m(m+n-2)))(2m+n-2)/(n-6)
(12)
gamma_2=(12(-16+44m+22m^2+20n-32mn+5m^2n-8n^2+5mn^2+n^3))/(m(m+n-2)(n-6)(n-8)).
(13)

The characteristic function can be computed, but it is rather messy and involves the generalized hypergeometric function _3F_2(a,b,c;d,e;z).

Letting

 w=((mF)/n)/(1+(mF)/n)
(14)

gives a beta distribution (Beyer 1987, p. 536).


See also

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

Explore with Wolfram|Alpha

References

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

Referenced on Wolfram|Alpha

Snedecor's F-Distribution

Cite this as:

Weisstein, Eric W. "Snedecor's F-Distribution." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/SnedecorsF-Distribution.html

Subject classifications