\HeaderA{Beta}{The Beta Distribution}{Beta}
\aliasA{dbeta}{Beta}{dbeta}
\aliasA{pbeta}{Beta}{pbeta}
\aliasA{qbeta}{Beta}{qbeta}
\aliasA{rbeta}{Beta}{rbeta}
\keyword{distribution}{Beta}
\begin{Description}\relax
Density, distribution function, quantile function and random
generation for the Beta distribution with parameters \code{shape1} and
\code{shape2} (and optional non-centrality parameter \code{ncp}).
\end{Description}
\begin{Usage}
\begin{verbatim}
dbeta(x, shape1, shape2, ncp = 0, log = FALSE)
pbeta(q, shape1, shape2, ncp = 0, lower.tail = TRUE, log.p = FALSE)
qbeta(p, shape1, shape2, ncp = 0, lower.tail = TRUE, log.p = FALSE)
rbeta(n, shape1, shape2, ncp = 0)
\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{shape1, shape2}] positive parameters of the Beta distribution.
\item[\code{ncp}] non-centrality parameter.
\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 Beta distribution with parameters \code{shape1} \eqn{= a}{} and
\code{shape2} \eqn{= b}{} has density
\deqn{f(x)=\frac{\Gamma(a+b)}{\Gamma(a)\Gamma(b)}{x}^{a} {(1-x)}^{b}
}{Gamma(a+b)/(Gamma(a)Gamma(b))x^(a-1)(1-x)^(b-1)}
for \eqn{a > 0}{}, \eqn{b > 0}{} and \eqn{0 \le x \le 1}{0 <= x <= 1}
where the boundary values at \eqn{x=0}{} or \eqn{x=1}{} are defined as
by continuity (as limits).
\\
The mean is \eqn{a/(a+b)}{} and the variance is \eqn{ab/((a+b)^2 (a+b+1))}{}.

\code{pbeta} is closely related to the incomplete beta function.  As
defined by Abramowitz and Stegun 6.6.1
\deqn{B_x(a,b) = \int_0^x t^{a-1} (1-t)^{b-1} dt,}{B_x(a,b) =
integral_0^x t^(a-1) (1-t)^(b-1) dt,}
and 6.6.2 \eqn{I_x(a,b) = B_x(a,b) / B(a,b)}{} where
\eqn{B(a,b) = B_1(a,b)}{} is the Beta function (\code{\LinkA{beta}{beta}}).

\eqn{I_x(a,b)}{} is \code{pbeta(x,a,b)}.

The noncentral Beta distribution (with \code{ncp} \eqn{ = \lambda}{})
is defined (Johnson et al, 1995, pp. 502) as the distribution of
\eqn{X/(X+Y)}{} where \eqn{X \sim \chi^2_{2a}(\lambda)}{X ~ chi^2_2a(lambda)}
and \eqn{Y \sim \chi^2_{2b}}{Y ~ chi^2_2b}.
\end{Details}
\begin{Value}
\code{dbeta} gives the density, \code{pbeta} the distribution
function, \code{qbeta} the quantile function, and \code{rbeta}
generates random deviates.

Invalid arguments will result in return value \code{NaN}, with a warning.
\end{Value}
\begin{Source}\relax
The central \code{dbeta} is based on a binomial probability, using code
contributed by Catherine Loader (see \code{\LinkA{dbinom}{dbinom}}) if either
shape parameter is larger than one, otherwise directly from the definition.
The non-central case is based on the derivation as a Poisson
mixture of betas (Johnson \emph{et al}, 1995, pp. 502--3).

The central \code{pbeta} uses a C translation (and enhancement for
\code{log\_p=TRUE}) of

Didonato, A. and Morris, A., Jr, (1992)
Algorithm 708: Significant digit computation of the incomplete beta
function ratios,
\emph{ACM Transactions on Mathematical Software}, \bold{18}, 360--373.
(See also\\ 
Brown, B. and Lawrence Levy, L. (1994)
Certification of algorithm 708: Significant digit computation of the
incomplete beta,
\emph{ACM Transactions on Mathematical Software}, \bold{20}, 393--397.)

The non-central \code{pbeta} uses a C translation of

Lenth, R. V. (1987) Algorithm AS226: Computing noncentral beta
probabilities. \emph{Appl. Statist}, \bold{36}, 241--244,
incorporating\\
Frick, H. (1990)'s AS R84, \emph{Appl. Statist}, \bold{39}, 311--2,
and\\
Lam, M.L. (1995)'s AS R95, \emph{Appl. Statist}, \bold{44}, 551--2.

\code{qbeta} is based on a C translation of

Cran, G. W., K. J. Martin and G. E. Thomas (1977).
Remark AS R19 and Algorithm AS 109,
\emph{Applied Statistics},  \bold{26}, 111--114,
and subsequent remarks (AS83 and correction).

\code{rbeta} is based on a C translation of

R. C. H. Cheng (1978).
Generating beta variates with nonintegral shape parameters.
\emph{Communications of the ACM}, \bold{21}, 317--322.
\end{Source}
\begin{References}\relax
Becker, R. A., Chambers, J. M. and Wilks, A. R. (1988)
\emph{The New S Language}.
Wadsworth \& Brooks/Cole.

Abramowitz, M. and Stegun, I. A. (1972)
\emph{Handbook of Mathematical Functions.} New York: Dover.
Chapter 6: Gamma and Related Functions.

Johnson, N. L., Kotz, S. and Balakrishnan, N. (1995)
\emph{Continuous Univariate Distributions}, volume 2, especially
chapter 25. Wiley, New York.
\end{References}
\begin{SeeAlso}\relax
\code{\LinkA{beta}{beta}} for the Beta function, and \code{\LinkA{dgamma}{dgamma}} for
the Gamma distribution.
\end{SeeAlso}
\begin{Examples}
\begin{ExampleCode}
x <- seq(0, 1, length=21)
dbeta(x, 1, 1)
pbeta(x, 1, 1)
\end{ExampleCode}
\end{Examples}

