Density, distribution function, quantile function and random
generation for the Beta distribution with parameters shape1
and
shape2
(and optional noncentrality parameter ncp
).
1 2 3 4 
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 P[X ≤ x], otherwise, P[X > x]. 
The Beta distribution with parameters shape1
= a and
shape2
= b has density
Γ(a+b)/(Γ(a)Γ(b))x^(a1)(1x)^(b1)
for a > 0, b > 0 and 0 ≤ x ≤ 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) = integral_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
= λ)
is defined (Johnson et al, 1995, pp. 502) as the distribution of
X/(X+Y) where X ~ chi^2_2a(λ)
and Y ~ 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 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.
Supplying ncp = 0
uses the algorithm for the noncentral
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.
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 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 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, 360–373.
(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.)
We have slightly tweaked the original “TOMS 708” algorithm, and
enhanced for log.p = TRUE
. For that (logscale) case,
underflow to Inf
(i.e., P = 0) or 0
, (i.e.,
P = 1) still happens because the original algorithm was designed
without logscale considerations. Underflow to Inf
now
typically signals a warning
.
The noncentral 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.
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.
Distributions for other standard distributions.
beta
for the Beta function.
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37  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 < 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 = 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

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