Beta
The Beta Distribution
Density, distribution function, quantile function and random
generation for the Beta distribution with parameters shape1 and
shape2 (and optional non-centrality parameter ncp).
- Keywords
- distribution
Usage
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)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. - shape1, shape2
- non-negative parameters of the Beta distribution.
- ncp
- non-centrality parameter.
- log, log.p
- logical; if TRUE, probabilities p are given as log(p).
- lower.tail
- logical; if TRUE (default), probabilities are $P[X \le x]$, otherwise, $P[X > x]$.
Details
The Beta distribution with parameters shape1 $= a$ and
shape2 $= b$ has density
$$f(x)=\frac{\Gamma(a+b)}{\Gamma(a)\Gamma(b)}{x}^{a-1} {(1-x)}^{b-1}%
$$
for $a > 0$, $b > 0$ 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
[dpqr]beta() functions are defined correspondingly.
pbeta is closely related to the incomplete beta function. As
defined by Abramowitz and Stegun 6.6.1
$$B_x(a,b) = \int_0^x t^{a-1} (1-t)^{b-1} dt,$$
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 (beta).
$I_x(a,b)$ is pbeta(x, a, b).
The noncentral Beta distribution (with ncp $ = \lambda$)
is defined (Johnson et al, 1995, pp.\ifelse{latex}{\out{~}}{ } 502) as the distribution of
$X/(X+Y)$ where $X ~ chi^2_2a(\lambda)$
and $Y ~ chi^2_2b$.
Value
dbeta gives the density, pbeta the distribution
function, qbeta the quantile function, and rbeta
generates random deviates.Invalid arguments will result in return value NaN, with a warning.The length of the result is determined by n for
rbeta, and is the maximum of the lengths of the
numerical arguments for the other functions.The numerical arguments other than n are recycled to the
length of the result. Only the first elements of the logical
arguments are used.
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.
Source
The central dbeta is based on a binomial probability, using code
contributed by Catherine Loader (see 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 et al, 1995, pp.\ifelse{latex}{\out{~}}{ } 502--3). The central pbeta uses a C translation (and enhancement for
log_p = TRUE) of 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--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--397.) The non-central pbeta uses a C translation of Lenth, R. V. (1987) Algorithm AS226: Computing noncentral beta
probabilities. Appl. Statist, 36, 241--244,
incorporating
Frick, H. (1990)'s AS R84, Appl. Statist, 39, 311--2,
and
Lam, M.L. (1995)'s AS R95, Appl. Statist, 44, 551--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. The central case of 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,
Applied Statistics, 26, 111--114,
and subsequent remarks (AS83 and correction). The central case of rbeta 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--322.
References
Becker, R. A., Chambers, J. M. and Wilks, A. R. (1988) The New S Language. Wadsworth & 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.
See Also
Distributions for other standard distributions.
beta for the Beta function.
Examples
library(stats)
x <- seq(0, 1, length = 21)
dbeta(x, 1, 1)
pbeta(x, 1, 1)
## Visualization, including limit cases:
pl.beta <- function(a,b, asp = if(isLim) 1, ylim = if(isLim) c(0,1.1)) {
if(isLim <- a == 0 || b == 0 || a == Inf || b == Inf) {
eps <- 1e-10
x <- c(0, eps, (1:7)/16, 1/2+c(-eps,0,eps), (9:15)/16, 1-eps, 1)
} else {
x <- seq(0, 1, length = 1025)
}
fx <- cbind(dbeta(x, a,b), pbeta(x, a,b), qbeta(x, a,b))
f <- fx; f[fx == Inf] <- 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