Beta {stats} | R Documentation |

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` |
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]. |

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)`

.

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(lambda)*
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 central `dbeta`

is based on a binomial probability, using code
contributed by Catherine Loader (see `dbinom`

) if either
shape parameter is larger than one, otherwise directly from the definition.
The non-central case is based on the derivation as a Poisson
mixture of betas (Johnson *et al*, 1995, pp. 502–3).

The central `pbeta`

uses a C translation (and enhancement for
`log_p=TRUE`

) of

Didonato, A. and Morris, A., Jr, (1992)
Algorithm 708: Significant digit computation of the incomplete beta
function ratios,
*ACM Transactions on Mathematical Software*, **18**, 360–373.
(See also

Brown, B. and Lawrence Levy, L. (1994)
Certification of algorithm 708: Significant digit computation of the
incomplete beta,
*ACM Transactions on Mathematical Software*, **20**, 393–397.)

The non-central `pbeta`

uses a C translation of

Lenth, R. V. (1987) Algorithm AS226: Computing noncentral beta
probabilities. *Appl. Statist*, **36**, 241–244,
incorporating

Frick, H. (1990)'s AS R84, *Appl. Statist*, **39**, 311–2,
and

Lam, M.L. (1995)'s AS R95, *Appl. Statist*, **44**, 551–2.

`qbeta`

is based on a C translation of

Cran, G. W., K. J. Martin and G. E. Thomas (1977).
Remark AS R19 and Algorithm AS 109,
*Applied Statistics*, **26**, 111–114,
and subsequent remarks (AS83 and correction).

`rbeta`

is based on a C translation of

R. C. H. Cheng (1978).
Generating beta variates with nonintegral shape parameters.
*Communications of the ACM*, **21**, 317–322.

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.

`beta`

for the Beta function, and `dgamma`

for
the Gamma distribution.

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

[Package *stats* version 2.9.0 Index]