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Beta {stats}

The Beta Distribution
R 3.0.2


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, ncp = 0, lower.tail = TRUE, log.p = FALSE)
rbeta(n, shape1, shape2, ncp = 0)


x, q
vector of quantiles.
vector of probabilities.
number of observations. If length(n) > 1, the length is taken to be the number required.
shape1, shape2
positive parameters of the Beta distribution.
non-centrality parameter.
log, log.p
logical; if TRUE, probabilities p are given as log(p).
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^(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 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 parameters for the other functions. The numerical parameters other than n are recycled to the length of the result. Only the first elements of the logical parameters are used.


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.


Supplying ncp = 0 uses the algorithm for the non-central 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.

See Also

Distributions for other standard distributions.

beta for the Beta function.


x <- seq(0, 1, length = 21)
dbeta(x, 1, 1)
pbeta(x, 1, 1)

Documentation reproduced from R 3.0.2. License: GPL-2.