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<table width="100%" summary="page for Beta {stats}"><tr><td>Beta {stats}</td><td style="text-align: right;">R Documentation</td></tr></table>

<h2>The Beta Distribution</h2>

<h3>Description</h3>

Density, distribution function, quantile function and random
generation for the Beta distribution with parameters <code>shape1</code> and
<code>shape2</code> (and optional non-centrality parameter <code>ncp</code>).

<h3>Usage</h3>

<pre>
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)
</pre>

<h3>Arguments</h3>

<table summary="R argblock">
<tr valign="top"><td><code>x, q</code></td>
<td>
vector of quantiles.
</td></tr>
<tr valign="top"><td><code>p</code></td>
<td>
vector of probabilities.
</td></tr>
<tr valign="top"><td><code>n</code></td>
<td>
number of observations. If <code>length(n) &gt; 1</code>, the length
is taken to be the number required.
</td></tr>
<tr valign="top"><td><code>shape1, shape2</code></td>
<td>
non-negative parameters of the Beta distribution.
</td></tr>
<tr valign="top"><td><code>ncp</code></td>
<td>
non-centrality parameter.
</td></tr>
<tr valign="top"><td><code>log, log.p</code></td>
<td>
logical; if TRUE, probabilities p are given as log(p).
</td></tr>
<tr valign="top"><td><code>lower.tail</code></td>
<td>
logical; if TRUE (default), probabilities are
P[X &le; x], otherwise, P[X &gt; x].
</td></tr>
</table>

<h3>Details</h3>

The Beta distribution with parameters <code>shape1</code> = a and
<code>shape2</code> = b has density

&Gamma;(a+b)/(&Gamma;(a)&Gamma;(b))x^(a-1)(1-x)^(b-1)

for a &gt; 0, b &gt; 0 and 0 &le; x &le; 1
where the boundary values at x=0 or x=1 are defined as
by continuity (as limits).
 
The mean is a/(a+b) and the variance is ab/((a+b)^2 (a+b+1)).
These moments and all distributional properties can be defined as
limits (leading to point masses at 0, 1/2, or 1) when a or
b are zero or infinite, and the corresponding
<code>[dpqr]beta()</code> functions are defined correspondingly.

<code>pbeta</code> is closely related to the incomplete beta function. As
defined by Abramowitz and Stegun 6.6.1

B_x(a,b) =
 integral_0^x t^(a-1) (1-t)^(b-1) dt,

and 6.6.2 I_x(a,b) = B_x(a,b) / B(a,b) where
B(a,b) = B_1(a,b) is the Beta function (<code><a href="../../base/html/Special.html">beta</a></code>).

I_x(a,b) is <code>pbeta(x, a, b)</code>.

The noncentral Beta distribution (with <code>ncp</code> = &lambda;)
is defined (Johnson et al, 1995, pp. 502) as the distribution of
X/(X+Y) where X ~ chi^2_2a(&lambda;)
and Y ~ chi^2_2b.

<h3>Value</h3>

<code>dbeta</code> gives the density, <code>pbeta</code> the distribution
function, <code>qbeta</code> the quantile function, and <code>rbeta</code>
generates random deviates.

Invalid arguments will result in return value <code>NaN</code>, with a warning.

The length of the result is determined by <code>n</code> for
<code>rbeta</code>, and is the maximum of the lengths of the
numerical arguments for the other functions.

The numerical arguments other than <code>n</code> are recycled to the
length of the result. Only the first elements of the logical
arguments are used.

Supplying <code>ncp = 0</code> uses the algorithm for the non-central
distribution, which is not the same algorithm used if <code>ncp</code> is
omitted. This is to give consistent behaviour in extreme cases with
values of <code>ncp</code> very near zero.

<h3>Source</h3>

<ul>
<li> The central <code>dbeta</code> is based on a binomial probability, using code
contributed by Catherine Loader (see <code><a href="Binomial.html">dbinom</a></code>) 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 et al, 1995, pp. 502&ndash;3).

</li>
<li> The central <code>pbeta</code> for the default (<code>log_p = FALSE</code>)
uses a C translation based on

Didonato, A. and Morris, A., Jr, (1992)
Algorithm 708: Significant digit computation of the incomplete beta
function ratios,
ACM Transactions on Mathematical Software, 18, 360&ndash;373.
(See also 
Brown, B. and Lawrence Levy, L. (1994)
Certification of algorithm 708: Significant digit computation of the
incomplete beta,
ACM Transactions on Mathematical Software, 20, 393&ndash;397.)
 
We have slightly tweaked the original &ldquo;TOMS 708&rdquo; algorithm, and
enhanced for <code>log.p = TRUE</code>. For that (log-scale) case,
underflow to <code>-Inf</code> (i.e., P = 0) or <code>0</code>, (i.e.,
P = 1) still happens because the original algorithm was designed
without log-scale considerations. Underflow to <code>-Inf</code> now
typically signals a <code><a href="../../base/html/warning.html">warning</a></code>.

</li>
<li> The non-central <code>pbeta</code> uses a C translation of

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

This computes the lower tail only, so the upper tail suffers from
cancellation and a warning will be given when this is likely to be
significant.

</li>
<li> The central case of <code>qbeta</code> 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,
Applied Statistics, 26, 111&ndash;114,
and subsequent remarks (AS83 and correction).

</li>
<li> The central case of <code>rbeta</code> is based on a C translation of

R. C. H. Cheng (1978).
Generating beta variates with nonintegral shape parameters.
Communications of the ACM, 21, 317&ndash;322.

</li></ul>

<h3>References</h3>

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

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

Johnson, N. L., Kotz, S. and Balakrishnan, N. (1995)
Continuous Univariate Distributions, volume 2, especially
chapter 25. Wiley, New York.

<a href="Distributions.html">Distributions</a> for other standard distributions.

<code><a href="../../base/html/Special.html">beta</a></code> for the Beta function.

<h3>Examples</h3>

<pre>
x &lt;- seq(0, 1, length = 21)
dbeta(x, 1, 1)
pbeta(x, 1, 1)

## Visualization, including limit cases:
pl.beta &lt;- function(a,b, asp = if(isLim) 1, ylim = if(isLim) c(0,1.1)) {
  if(isLim &lt;- a == 0 || b == 0 || a == Inf || b == Inf) {
    eps &lt;- 1e-10
    x &lt;- c(0, eps, (1:7)/16, 1/2+c(-eps,0,eps), (9:15)/16, 1-eps, 1)
  } else {
    x &lt;- seq(0, 1, length = 1025)
  }
  fx &lt;- cbind(dbeta(x, a,b), pbeta(x, a,b), qbeta(x, a,b))
  f &lt;- fx; f[fx == Inf] &lt;- 1e100
  matplot(x, f, ylab="", type="l", ylim=ylim, asp=asp,
          main = sprintf("[dpq]beta(x, a=%g, b=%g)", a,b))
  abline(0,1,     col="gray", lty=3)
  abline(h = 0:1, col="gray", lty=3)
  legend("top", paste0(c("d","p","q"), "beta(x, a,b)"),
         col=1:3, lty=1:3, bty = "n")
  invisible(cbind(x, fx))
}
pl.beta(3,1)

pl.beta(2, 4)
pl.beta(3, 7)
pl.beta(3, 7, asp=1)

pl.beta(0, 0)   ## point masses at  {0, 1}

pl.beta(0, 2)   ## point mass at 0 ; the same as
pl.beta(1, Inf)

pl.beta(Inf, 2) ## point mass at 1 ; the same as
pl.beta(3, 0)

pl.beta(Inf, Inf)# point mass at 1/2
</pre>

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