\Header{TDist}{The Student t Distribution}
\alias{dt}{TDist}
\alias{pt}{TDist}
\alias{qt}{TDist}
\alias{rt}{TDist}
\keyword{distribution}{TDist}
\begin{Description}\relax
Density, distribution function, quantile function and random
generation for the t distribution with \code{df} degrees of freedom
(and optional noncentrality parameter \code{ncp}).
\end{Description}
\begin{Usage}
\begin{verbatim}
dt(x, df, ncp=0, log = FALSE)
pt(q, df, ncp=0, lower.tail = TRUE, log.p = FALSE)
qt(p, df,        lower.tail = TRUE, log.p = FALSE)
rt(n, df)
\end{verbatim}
\end{Usage}
\begin{Arguments}
\begin{ldescription}
\item[\code{x, q}] vector of quantiles.
\item[\code{p}] vector of probabilities.
\item[\code{n}] number of observations. If \code{length(n) > 1}, the length
is taken to be the number required.
\item[\code{df}] degrees of freedom (\eqn{> 0}{}, maybe non-integer).
\item[\code{ncp}] non-centrality parameter \eqn{\delta}{};
currently for \code{pt()} and \code{dt()}, only for \code{ncp <= 37.62}.
\item[\code{log, log.p}] logical; if TRUE, probabilities p are given as log(p).
\item[\code{lower.tail}] logical; if TRUE (default), probabilities are
\eqn{P[X \le x]}{P[X <= x]}, otherwise, \eqn{P[X > x]}{P[X > x]}.
\end{ldescription}
\end{Arguments}
\begin{Details}\relax
The \eqn{t}{} distribution with \code{df} \eqn{= \nu}{= n} degrees of
freedom has density
\deqn{
f(x) = \frac{\Gamma ((\nu+1)/2)}{\sqrt{\pi \nu} \Gamma (\nu/2)}
(1 + x^2/\nu)^{-(\nu+1)/2}
}{f(x) = Gamma((n+1)/2) / (sqrt(n pi) Gamma(n/2)) (1 + x^2/n)^-((n+1)/2)}
for all real \eqn{x}{}.
It has mean \eqn{0}{} (for \eqn{\nu > 1}{n > 1}) and
variance \eqn{\frac{\nu}{\nu-2}}{n/(n-2)} (for \eqn{\nu > 2}{n > 2}).

The general \emph{non-central} \eqn{t}{}
with parameters \eqn{(\nu,\delta)}{(df,Del)} \code{= (df, ncp)}
is defined as a the distribution of
\eqn{T_{\nu}(\delta) := \frac{U + \delta}{\chi_{\nu}/\sqrt{\nu}}}{T(df,Del) := (U + Del) / (Chi(df) / sqrt(df)) }
where \eqn{U}{} and \eqn{\chi_{\nu}}{Chi(df)}  are independent random
variables, \eqn{U \sim {\cal N}(0,1)}{U \~ N(0,1)}, and
\eqn{\chi^2_\nu}{Chi(df)^2}
is chi-squared, see \code{\Link{pchisq}}.

The most used applications are power calculations for \eqn{t}{}-tests:\\
Let \eqn{T= \frac{\bar{X} - \mu_0}{S/\sqrt{n}}}{T= (mX - m0) / (S/sqrt(n))}
where
\eqn{\bar{X}}{mX} is the \code{\Link{mean}} and \eqn{S}{} the sample standard
deviation (\code{\Link{sd}}) of \eqn{X_1,X_2,\dots,X_n}{} which are i.i.d.
\eqn{       N(\mu,\sigma^2)}{N(mu,sigma^2)}.
Then \eqn{T}{} is distributed as non-centrally \eqn{t}{} with
\code{df}\eqn{= n-1}{}
degrees of freedom and \bold{n}on-\bold{c}entrality \bold{p}arameter
\code{ncp}\eqn{= (\mu - \mu_0) \sqrt{n}/\sigma}{= (mu - m0) * sqrt(n)/sigma}.
\end{Details}
\begin{Value}
\code{dt} gives the density,
\code{pt} gives the distribution function,
\code{qt} gives the quantile function, and
\code{rt} generates random deviates.
\end{Value}
\begin{References}\relax
Becker, R. A., Chambers, J. M. and Wilks, A. R. (1988)
\emph{The New S Language}.
Wadsworth \& Brooks/Cole. (except non-central versions.)

Lenth, R. V. (1989). \emph{Algorithm AS 243} ---
Cumulative distribution function of the non-central \eqn{t}{} distribution,
\emph{Appl.\ Statist.} \bold{38}, 185--189.
\end{References}
\begin{SeeAlso}\relax
\code{\Link{df}} for the F distribution.
\end{SeeAlso}
\begin{Examples}
\begin{ExampleCode}
1 - pt(1:5, df = 1)
qt(.975, df = c(1:10,20,50,100,1000))

tt <- seq(0,10, len=21)
ncp <- seq(0,6, len=31)
ptn <- outer(tt,ncp, function(t,d) pt(t, df = 3, ncp=d))
image(tt,ncp,ptn, zlim=c(0,1),main=t.tit <- "Non-central t - Probabilities")
persp(tt,ncp,ptn, zlim=0:1, r=2, phi=20, theta=200, main=t.tit,
      xlab = "t", ylab = "noncentrality parameter", zlab = "Pr(T <= t)")

op <- par(yaxs="i")
plot(function(x) dt(x, df = 3, ncp = 2), -3, 11, ylim = c(0, 0.32),
     main="Non-central t - Density")
par(op)
\end{ExampleCode}
\end{Examples}

