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 noncentrality 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 
shape1 , shape2 
nonnegative parameters of the Beta distribution. 
ncp 
noncentrality parameter. 
log , log.p 
logical; if TRUE, probabilities p are given as log(p). 
lower.tail 
logical; if TRUE (default), probabilities are

Details
The Beta distribution with parameters shape1
= a
and
shape2
= b
has density
f(x)=\frac{\Gamma(a+b)}{\Gamma(a)\Gamma(b)}{x}^{a1} {(1x)}^{b1}%
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))
.
If a,b > 1
, (or one of them =1
), the mode is (a1)/(a+b2)
.
These and all other 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^{a1} (1t)^{b1} 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}
.
There, \chi^2_n(\lambda)
is the noncentral
chisquared distribution with n
degrees of freedom and
noncentrality parameter \lambda
, see Chisquare.
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 noncentral
distribution, which is not the same algorithm as when 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 (seedbinom
) if either shape parameter is larger than one, otherwise directly from the definition. The noncentral case is based on the derivation as a Poisson mixture of betas (Johnson et al., 1995, pp. 502–3).The central
pbeta
for the default (log_p = FALSE
) uses a C translation based onDidonato, 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, doi:10.1145/131766.131776. (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, doi:10.1145/192115.192155.)
We have slightly tweaked the original “TOMS 708” algorithm, and enhanced forlog.p = TRUE
. For that (logscale) case, underflow toInf
(i.e.,P = 0
) or0
, (i.e.,P = 1
) still happens because the original algorithm was designed without logscale considerations. Underflow toInf
now typically signals awarning
.The noncentral
pbeta
uses a C translation ofLenth, R. V. (1987) Algorithm AS 226: Computing noncentral beta probabilities. Applied Statistics, 36, 241–244, doi:10.2307/2347558, incorporating
Frick, H. (1990)'s AS R84, Applied Statistics, 39, 311–2, doi:10.2307/2347780 and
Lam, M.L. (1995)'s AS R95, Applied Statistics, 44, 551–2, doi:10.2307/2986147.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 ofCran, G. W., K. J. Martin and G. E. Thomas (1977). Remark AS R19 and Algorithm AS 109, Applied Statistics, 26, 111–114, doi:10.2307/2346887, and subsequent remarks (AS83 and correction).
Enhancements, notably for starting values and switching to a logscale Newton search, by R Core.
The central case of
rbeta
is based on a C translation ofR. 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
x < seq(0, 1, length.out = 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 < 1e10
x < c(0, eps, (1:7)/16, 1/2+c(eps,0,eps), (9:15)/16, 1eps, 1)
} else {
x < seq(0, 1, length.out = 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