| Beta {stats} | R Documentation |
Density, distribution function, quantile function and random
generation for the Beta distribution with parameters shape1 and
shape2 (and optional non-centrality parameter ncp).
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)
x, q |
vector of quantiles. |
p |
vector of probabilities. |
n |
number of observations. If |
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
|
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}.
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 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.
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.
beta for the Beta function, and dgamma for
the Gamma distribution.
x <- seq(0, 1, length=21)
dbeta(x, 1, 1)
pbeta(x, 1, 1)