# FDist {stats}

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

### 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.
`Inf`

is allowed. - ncp
- non-centrality parameter. If omitted the central F is assumed.
- 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 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.

### Values

`df`

gives the density, `pf`

gives the distribution function `qf`

gives the quantile function, and `rf`

generates random deviates.

Invalid arguments will result in return value `NaN`

, with a warning.

The length of the result is determined by `n`

for `rf`

, and is the maximum of the lengths of the numerical parameters for the other functions.

The numerical parameters other than `n`

are recycled to the length of the result. Only the first elements of the logical parameters are used.

### References

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

Johnson, N. L., Kotz, S. and Balakrishnan, N. (1995) *Continuous Univariate Distributions*, volume 2, chapters 27 and 30. Wiley, New York.

### Note

Supplying `ncp = 0`

uses the algorithm for the non-central distribution, which is not the same algorithm used if `ncp`

is omitted. This is to give consistent behaviour in extreme cases with values of `ncp`

very near zero.

The code for non-zero `ncp`

is principally intended to be used for moderate values of `ncp`

: it will not be highly accurate, especially in the tails, for large values.

### See Also

Distributions for other standard distributions, including `dchisq`

for chi-squared and `dt`

for Student's t distributions.

### Examples

## Equivalence of pt(.,nu) with pf(.^2, 1,nu): x <- seq(0.001, 5, len = 100) nu <- 4 stopifnot(all.equal(2*pt(x,nu) - 1, pf(x^2, 1,nu)), ## upper tails: all.equal(2*pt(x, nu, lower=FALSE), pf(x^2, 1,nu, lower=FALSE))) ## 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} 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

Documentation reproduced from R 3.0.2. License: GPL-2.