| 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, lower.tail = TRUE, log.p = FALSE)
rbeta(n, shape1, shape2)
Arguments
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
|
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).
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).
Value
dbeta gives the density, pbeta the distribution
function, qbeta the quantile function, and rbeta
generates random deviates.
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.
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)