Fd-class

Class "Fd"

The F distribution with df1 = $n_1$, by default = 1, and df2 = $n_2$, by default = 1, degrees of freedom has density $$d(x) = \frac{\Gamma(n_1/2 + n_2/2)}{\Gamma(n_1/2)\Gamma(n_2/2)} \left(\frac{n_1}{n_2}\right)^{n_1/2} x^{n_1/2 -1} \left(1 + \frac{n_1 x}{n_2}\right)^{-(n_1 + n_2) / 2}$$ for $x > 0$. C.f. rf

Note

It is the distribution of the ratio of the mean squares of n1 and n2 independent standard normals, and hence of the ratio of two independent chi-squared variates each divided by its degrees of freedom. Since the ratio of a normal and the root mean-square of m independent normals has a Student's $t_m$ distribution, the square of a $t_m$ variate has a F distribution on 1 and m degrees of freedom. The non-central F distribution is again the ratio of mean squares of independent normals of unit variance, but those in the numerator are allowed to have non-zero means and ncp is the sum of squares of the means.

Objects from the Class

Objects can be created by calls of the form Fd(df1, df2). This object is a F distribution.

Extends

Class "AbscontDistribution", directly. Class "UnivariateDistribution", by class "AbscontDistribution". Class "Distribution", by class "AbscontDistribution".

Ad hoc methods

• An ad hoc method is provided for slotdifncp!=0.
• For R Version<2.3.0< code="">ad hoc methods are provided for slotsq,rifncp!=0; for R Version>=2.3.0the methods from packagestatsare used.

concept

• F distribution
• absolutely continuous distribution
• S4 distribution class
• generating function

See Also

FParameter-class AbscontDistribution-class Reals-class rf

Aliases

• Fd-class
• Fd
• initialize,Fd-method

Examples

F <- Fd(df1 = 1, df2 = 1) # F is a F distribution with df=1 and df2=1.
r(F)(1) # one random number generated from this distribution, e.g. 29.37863
d(F)(1) # Density of this distribution is 0.1591549 for x=1 .
p(F)(1) # Probability that x<1 is 0.5.
q(F)(.1) # Probability that x<0.02508563 is 0.1.
df1(F) # df1 of this distribution is 1.
df1(F) <- 2 # df1 of this distribution is now 2.
Fn <- Fd(df1 = 1, df2 = 1, ncp = 0.5) 
  # Fn is a F distribution with df=1, df2=1 and ncp =0.5.
d(Fn)(1) ## from R 2.3.0 on ncp no longer ignored...