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  \code{pf} gives the distribution function
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  \code{pf} gives the distribution function
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  \code{qf} gives the quantile function, and
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  \code{qf} gives the quantile function, and
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  \code{rf} generates random deviates.
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  \code{rf} generates random deviates.
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  Invalid arguments will result in return value \code{NaN}, with a warning.
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  Invalid arguments will result in return value \code{NaN}, with a warning.
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  The length of the result is determined by \code{n} for
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  The length of the result is determined by \code{n} for
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  \code{rf}, and is the maximum of the lengths of the
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  \code{rf}, and is the maximum of the lengths of the
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  numerical parameters for the other functions.
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  numerical parameters for the other functions.
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  The numerical parameters other than \code{n} are recycled to the
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  The numerical parameters other than \code{n} are recycled to the
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  length of the result.  Only the first elements of the logical
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  length of the result.  Only the first elements of the logical
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  parameters are used.
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  parameters are used.
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}
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}
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\details{
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\details{
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  \link{Distributions} for other standard distributions, including
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  \link{Distributions} for other standard distributions, including
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  \code{\link{dchisq}} for chi-squared and \code{\link{dt}} for Student's
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  \code{\link{dchisq}} for chi-squared and \code{\link{dt}} for Student's
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  t distributions.
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  t distributions.
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}
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}
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\examples{
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\examples{
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## Equivalence of pt(.,nu) with pf(.^2, 1,nu):
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x <- seq(0.001, 5, len = 100)
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nu <- 4
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stopifnot(all.equal(2*pt(x,nu) - 1, pf(x^2, 1,nu)),
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          ## upper tails:
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 	  all.equal(2*pt(x,     nu, lower=FALSE),
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		      pf(x^2, 1,nu, lower=FALSE)))
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## the density of the square of a t_m is 2*dt(x, m)/(2*x)
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## the density of the square of a t_m is 2*dt(x, m)/(2*x)
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# check this is the same as the density of F_{1,m}
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# check this is the same as the density of F_{1,m}
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x <- seq(0.001, 5, len = 100)
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all.equal(df(x^2, 1, 5), dt(x, 5)/x)
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all.equal(df(x^2, 1, 5), dt(x, 5)/x)
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## Identity:  qf(2*p - 1, 1, df)) == qt(p, df)^2)  for  p >= 1/2
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## Identity:  qf(2*p - 1, 1, df)) == qt(p, df)^2)  for  p >= 1/2
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p <- seq(1/2, .99, length = 50); df <- 10
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p <- seq(1/2, .99, length = 50); df <- 10
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rel.err <- function(x, y) ifelse(x == y, 0, abs(x-y)/mean(abs(c(x,y))))
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rel.err <- function(x, y) ifelse(x == y, 0, abs(x-y)/mean(abs(c(x,y))))