Beta {stats}R Documentation

The Beta Distribution


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


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.

See Also

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)

[Package stats version 2.2.1 Index]