52.2.6 F Random Variable

Let \(S_1\) and \(S_2\) be independent random variables with a \(\chi^2\) distribution with degrees of freedom \(n\) and \(m\), respectively. Then

has an \(F\) distribution with \(n\) and \(m\) degrees of freedom.

Function: pdf_f (x,m,n) ¶

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

\[f(x; m, n) = \cases{ B\left(\displaystyle{m\over 2}, \displaystyle{n\over 2}\right)^{-1} \left(\displaystyle{m\over n}\right)^{m/ 2} x^{m/2-1} \left(1 + \displaystyle{m\over n}x\right)^{-\left(n+m\right)/2} & $x > 0$ \cr \cr 0 & otherwise } \]

Categories: Package distrib ·

Function: cdf_f (x,m,n) ¶

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

Categories: Package distrib ·

Function: quantile_f (q,m,n) ¶

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

Categories: Package distrib ·

Function: mean_f (m,n) ¶

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

\[E[X] = {n\over n-2} \]

Categories: Package distrib ·

Function: var_f (m,n) ¶

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

\[V[X] = {2n^2(n+m-2) \over m(n-4)(n-2)^2} \]

Categories: Package distrib ·

Function: std_f (m,n) ¶

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

\[D[X] = {\sqrt{2}\, n \over n-2} \sqrt{n+m-2\over m(n-4)} \]

Categories: Package distrib ·

Function: skewness_f (m,n) ¶

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

\[SK[X] = {(n+2m-2)\sqrt{8(n-4)} \over (n-6)\sqrt{m(n+m-2)}} \]

Categories: Package distrib ·

Function: kurtosis_f (m,n) ¶

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

\[KU[X] = 12{m(n+m-2)(5n-22) + (n-4)(n-2)^2 \over m(n-8)(n-6)(n+m-2)} \]

Categories: Package distrib ·

Function: random_f (m,n)
    random_f (m,n,k) ¶

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").

Categories: Package distrib · Random numbers ·