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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
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
Returns the value at x of the distribution function of a F random variable \(F(m,n)\), with \(m,n>0\).
The cdf is
where
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
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
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
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
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
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
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")
.