extraDistr (version 1.10.0)

BetaPrime: Beta prime distribution

Description

Density, distribution function, quantile function and random generation for the beta prime distribution.

Usage

dbetapr(x, shape1, shape2, scale = 1, log = FALSE)
pbetapr(q, shape1, shape2, scale = 1, lower.tail = TRUE, log.p = FALSE)
qbetapr(p, shape1, shape2, scale = 1, lower.tail = TRUE, log.p = FALSE)
rbetapr(n, shape1, shape2, scale = 1)

Arguments

x, q

vector of quantiles.

shape1, shape2

non-negative parameters.

scale

positive valued scale 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, \(P[X > x]\).

p

vector of probabilities.

n

number of observations. If length(n) > 1, the length is taken to be the number required.

Details

If \(X \sim \mathrm{Beta}(\alpha, \beta)\), then \(\frac{X}{1-X} \sim \mathrm{BetaPrime}(\alpha, \beta)\).

Probability density function

$$ f(x) = \frac{(x/\sigma)^{\alpha-1} (1+x/\sigma)^{-\alpha -\beta}}{\mathrm{B}(\alpha,\beta)\sigma} $$

Cumulative distribution function

$$ F(x) = I_{\frac{x/\sigma}{1+x/\sigma}}(\alpha, \beta) $$

See Also

Beta

Examples

x <- rbetapr(1e5, 5, 3, 2)
hist(x, 350, freq = FALSE, xlim = c(0, 100))
curve(dbetapr(x, 5, 3, 2), 0, 100, col = "red", add = TRUE, n = 500)
hist(pbetapr(x, 5, 3, 2))
plot(ecdf(x), xlim = c(0, 100))
curve(pbetapr(x, 5, 3, 2), 0, 100, col = "red", add = TRUE, n = 500)