# Beta {stats}

### Description

Density, distribution function, quantile function and random generation for the Beta distribution with parameters `shape1`

and `shape2`

(and optional non-centrality parameter `ncp`

).

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

### 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

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.

### Values

`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.

### References

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.

### Note

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.

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