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#### 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 $$ F = {S_1/n \over S_2/m} $$ 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 ·

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