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Let S_1 and S_2 be independent random variables with a \(\chi^2\) distribution with degrees of freedom n and m, respectively. Then $$ F = {S_1/n \over S_2/m} $$ has an F distribution with n and m degrees of freedom.
Returns the value at x of the density function of a F random variable F(m,n), with m,n>0. To make use of this function, write first load("distrib").
The pdf is $$ f(x; m, n) = \cases{ B\left(\displaystyle{m\over 2}, \displaystyle{n\over 2}\right)^{-1} \left(\displaystyle{m\over n}\right)^{m/ 2} x^{m/2-1} \left(1 + \displaystyle{m\over n}x\right)^{-\left(n+m\right)/2} & $x > 0$ \cr \cr 0 & otherwise } $$
Returns the value at x of the distribution function of a F random variable F(m,n), with m,n>0.
The cdf is $$ F(x; m, n) = \cases{ 1 - I_z\left(\displaystyle{m\over 2}, {n\over 2}\right) & $x > 0$ \cr 0 & otherwise } $$
where $$ z = {n\over mx+n} $$
and \(I_z(a,b)\) is the beta_incomplete_regularized function.
(%i1) load ("distrib")$
(%i2) cdf_f(2,3,9/4);
9 3 3
(%o2) 1 - beta_incomplete_regularized(-, -, --)
8 2 11
(%i3) float(%); (%o3) 0.6675672817900802
Returns the q-quantile of a F random variable F(m,n), with m,n>0; in other words, this is the inverse of cdf_f. Argument q must be an element of [0,1].
(%i1) load ("distrib")$
(%i2) quantile_f(2/5,sqrt(3),5); (%o2) 0.5189478385736904
Returns the mean of a F random variable F(m,n), with m>0, n>2. To make use of this function, write first load("distrib").
The mean is $$ E[X] = {n\over n-2} $$
Returns the variance of a F random variable F(m,n), with m>0, n>4. To make use of this function, write first load("distrib").
The variance is $$ V[X] = {2n^2(n+m-2) \over m(n-4)(n-2)^2} $$
Returns the standard deviation of a F random variable F(m,n), with m>0, n>4. To make use of this function, write first load("distrib").
The standard deviation is $$ D[X] = {\sqrt{2}\, n \over n-2} \sqrt{n+m-2\over m(n-4)} $$
Returns the skewness coefficient of a F random variable F(m,n), with m>0, n>6. To make use of this function, write first load("distrib").
The skewness coefficient is $$ SK[X] = {(n+2m-2)\sqrt{8(n-4)} \over (n-6)\sqrt{m(n+m-2)}} $$
Returns the kurtosis coefficient of a F random variable F(m,n), with m>0, n>8. To make use of this function, write first load("distrib").
The kurtosis coefficient is $$ KU[X] = 12{m(n+m-2)(5n-22) + (n-4)(n-2)^2 \over m(n-8)(n-6)(n+m-2)} $$
Returns a F random variate F(m,n), with m,n>0. Calling random_f with a third argument k, a random sample of size k will be simulated.
The simulation algorithm is based on the fact that if X is a Chi^2(m) random variable and Y is a \(\chi^2(n)\) random variable, then $$ F={{n X}\over{m Y}} $$
is a F random variable with m and n degrees of freedom, F(m,n).
To make use of this function, write first load("distrib").