FDist {stats} | R Documentation |

## The F Distribution

### Description

Density, distribution function, quantile function and random
generation for the F distribution with `df1`

and `df2`

degrees of freedom (and optional non-centrality parameter
`ncp`

).

### Usage

df(x, df1, df2, log = FALSE)
pf(q, df1, df2, ncp=0, lower.tail = TRUE, log.p = FALSE)
qf(p, df1, df2, lower.tail = TRUE, log.p = FALSE)
rf(n, df1, df2)

### Arguments

`x, q` |
vector of quantiles. |

`p` |
vector of probabilities. |

`n` |
number of observations. If `length(n) > 1` , the length
is taken to be the number required. |

`df1, df2` |
degrees of freedom. |

`ncp` |
non-centrality parameter. |

`log, log.p` |
logical; if TRUE, probabilities p are given as log(p). |

`lower.tail` |
logical; if TRUE (default), probabilities are
*P[X <= x]*, otherwise, *P[X > x]*. |

### Details

The F distribution with `df1 =`

*n1* and `df2 =`

*n2* degrees of freedom has density

*f(x) = Gamma((n1 + n2)/2) / (Gamma(n1/2) Gamma(n2/2))
(n1/n2)^(n1/2) x^(n1/2 - 1)
(1 + (n1/n2) x)^-(n1 + n2)/2*

for *x > 0*.

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. See `Chisquare`

for further details on
non-central distributions.

### Value

`df`

gives the density,
`pf`

gives the distribution function
`qf`

gives the quantile function, and
`rf`

generates random deviates.

### References

Becker, R. A., Chambers, J. M. and Wilks, A. R. (1988)
*The New S Language*.
Wadsworth & Brooks/Cole.

### See Also

`dchisq`

for chi-squared and `dt`

for Student's
t distributions.

### Examples

## the density of the square of a t_m is 2*dt(x, m)/(2*x)
# check this is the same as the density of F_{1,m}
x <- seq(0.001, 5, len=100)
all.equal(df(x^2, 1, 5), dt(x, 5)/x)
## Identity: qf(2*p - 1, 1, df)) == qt(p, df)^2) for p >= 1/2
p <- seq(1/2, .99, length=50); df <- 10
rel.err <- function(x,y) ifelse(x==y,0, abs(x-y)/mean(abs(c(x,y))))
quantile(rel.err(qf(2*p - 1, df1=1, df2=df), qt(p, df)^2), .90)# ~= 7e-9

[Package

*stats* version 2.1.0

Index]