Beta {stats} 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`).

### 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

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 non-central Beta distribution 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.

### Value

`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.

### Source

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 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 AS R84 (1990), Appl. Statist, 39, 311–2.

`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.

### 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.

`beta` for the Beta function, and `dgamma` for the Gamma distribution.
```x <- seq(0, 1, length=21)