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.

p

vector of probabilities.

n

number of observations. If `length(n) > 1`

, the length
is taken to be the number required.

shape1, shape2

non-negative 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 \(P[X \le x]\), otherwise, \(P[X > x]\).

`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 arguments for the other functions.

The numerical arguments other than `n`

are recycled to the
length of the result. Only the first elements of the logical
arguments are used.

The Beta distribution with parameters `shape1`

\(= a\) and
`shape2`

\(= b\) has density
$$f(x)=\frac{\Gamma(a+b)}{\Gamma(a)\Gamma(b)}{x}^{a-1} {(1-x)}^{b-1}%
$$
for \(a > 0\), \(b > 0\) and \(0 \le x \le 1\)
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 \(ab/((a+b)^2 (a+b+1))\).
These moments and all distributional properties can be defined as
limits (leading to point masses at 0, 1/2, or 1) when \(a\) or
\(b\) are zero or infinite, and the corresponding
`[dpqr]beta()`

functions are defined correspondingly.

`pbeta`

is closely related to the incomplete beta function. As
defined by Abramowitz and Stegun 6.6.1
$$B_x(a,b) = \int_0^x t^{a-1} (1-t)^{b-1} dt,$$
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)`

.

The noncentral Beta distribution (with `ncp`

\( = \lambda\))
is defined (Johnson *et al*, 1995, pp.502) as the distribution of
\(X/(X+Y)\) where \(X \sim \chi^2_{2a}(\lambda)\)
and \(Y \sim \chi^2_{2b}\).

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.

Distributions for other standard distributions.

`beta`

for the Beta function.

# NOT RUN { x <- seq(0, 1, length = 21) dbeta(x, 1, 1) pbeta(x, 1, 1) ## Visualization, including limit cases: pl.beta <- function(a,b, asp = if(isLim) 1, ylim = if(isLim) c(0,1.1)) { if(isLim <- a == 0 || b == 0 || a == Inf || b == Inf) { eps <- 1e-10 x <- c(0, eps, (1:7)/16, 1/2+c(-eps,0,eps), (9:15)/16, 1-eps, 1) } else { x <- seq(0, 1, length = 1025) } fx <- cbind(dbeta(x, a,b), pbeta(x, a,b), qbeta(x, a,b)) f <- fx; f[fx == Inf] <- 1e100 matplot(x, f, ylab="", type="l", ylim=ylim, asp=asp, main = sprintf("[dpq]beta(x, a=%g, b=%g)", a,b)) abline(0,1, col="gray", lty=3) abline(h = 0:1, col="gray", lty=3) legend("top", paste0(c("d","p","q"), "beta(x, a,b)"), col=1:3, lty=1:3, bty = "n") invisible(cbind(x, fx)) } pl.beta(3,1) pl.beta(2, 4) pl.beta(3, 7) pl.beta(3, 7, asp=1) pl.beta(0, 0) ## point masses at {0, 1} pl.beta(0, 2) ## point mass at 0 ; the same as pl.beta(1, Inf) pl.beta(Inf, 2) ## point mass at 1 ; the same as pl.beta(3, 0) pl.beta(Inf, Inf)# point mass at 1/2 # }