This help topic is for R version 2.9.0. For the current version of R, try https://stat.ethz.ch/R-manual/R-patched/library/stats/html/Beta.html
Beta {stats}R Documentation

The Beta Distribution

Description

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

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

positive 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-x)}^{b}%

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)).

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. 502) as the distribution of X/(X+Y) where X \sim \chi^2_{2a}(\lambda) and Y \sim \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.

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. 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.

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).

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

beta for the Beta function, and dgamma for the Gamma distribution.

Examples

x <- seq(0, 1, length=21)
dbeta(x, 1, 1)
pbeta(x, 1, 1)

[Package stats version 2.9.0 ]