# Fd-class

0th

Percentile

##### 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

Keywords
distribution
##### 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.

##### Slots

img

Object of class "Reals": The space of the image of this distribution has got dimension 1 and the name "Real Space".

param

Object of class "FParameter": the parameter of this distribution (df1 and df2), declared at its instantiation

r

Object of class "function": generates random numbers (calls function rf)

d

Object of class "function": density function (calls function df)

p

Object of class "function": cumulative function (calls function pf)

q

Object of class "function": inverse of the cumulative function (calls function qf)

.withArith

logical: used internally to issue warnings as to interpretation of arithmetics

.withSim

logical: used internally to issue warnings as to accuracy

.logExact

logical: used internally to flag the case where there are explicit formulae for the log version of density, cdf, and quantile function

.lowerExact

logical: used internally to flag the case where there are explicit formulae for the lower tail version of cdf and quantile function

Symmetry

object of class "DistributionSymmetry"; used internally to avoid unnecessary calculations.

##### Extends

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

##### Methods

initialize

signature(.Object = "Fd"): initialize method

df1

signature(object = "Fd"): returns the slot df1 of the parameter of the distribution

df1<-

signature(object = "Fd"): modifies the slot df1 of the parameter of the distribution

df2

signature(object = "Fd"): returns the slot df2 of the parameter of the distribution

df2<-

signature(object = "Fd"): modifies the slot df2 of the parameter of the distribution

• An ad hoc method is provided for slot d if ncp!=0.

• For R Version <2.3.0 ad hoc methods are provided for slots q, r if ncp!=0; for R Version >=2.3.0 the methods from package stats are used.

FParameter-class AbscontDistribution-class Reals-class rf

##### Aliases
• Fd-class
• Fd
• initialize,Fd-method
##### Examples
# NOT RUN {
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.
## in RStudio or Jupyter IRKernel, use q.l(.)(.) instead of q(.)(.)
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...
# }

Documentation reproduced from package distr, version 2.7.0, License: LGPL-3

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