| 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, lower.tail = TRUE, log.p = FALSE) rbeta(n, shape1, shape2)
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 <= x], otherwise, P[X > x]. |
The Beta distribution with parameters shape1 = a and
shape2 = b has density
Gamma(a+b)/(Gamma(a)Gamma(b))x^(a-1)(1-x)^(b-1)
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)).
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^(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).
dbeta gives the density, pbeta the distribution
function, qbeta the quantile function, and rbeta
generates random deviates.
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.
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)