# 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`

- 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

##### Ad hoc methods

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.

##### See Also

##### 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 2.8.0, License: LGPL-3*