Density, distribution function, quantile function and random generation for the Beta distribution with parameters
shape2 (and optional non-centrality parameter
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(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.
Invalid arguments will result in return value
NaN, with a warning. The length of the result is determined by
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.
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.
Distributions for other standard distributions.
beta for the Beta function.
Documentation reproduced from R 3.0.2. License: GPL-2.