beta (stats) {QRMlib}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).

Arguments

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].

Details

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);

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).

Value

dbeta gives the density, pbeta the distribution function, qbeta the quantile function, and rbeta generates random deviates.

Author(s)

documentation by Scott Ulman for R-language distribution

References

Becker, R. A., Chambers, J. M. and Wilks, A. R. (1988) The New S Language. Wadsworth, Brooks, and Cole.

Abramowitz, M. and Stegun, I. A. (1972) Handbook of Mathematical Functions. New York: Dover. Chapter 6: Gamma and Related Functions.

Examples

x <- seq(0, 1, length=21);
dbeta(x, 1, 1); #actually a standard uniform density
pbeta(x, 1, 1)  #actually a standard uniform distribution

[Package QRMlib version 1.4.4 Index]