extraDistr (version 1.8.1)

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 $X ~ Beta(\alpha, \beta)$, then $X/(1-X) ~ BetaPrime(\alpha, \beta)$.

Probability density function

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

Cumulative distribution function

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

See Also

Beta

Examples

x <- rbetapr(1e5, 5, 3, 2)
xx <- seq(0, 100, by = 0.1)
hist(x, 350, freq = FALSE, xlim = c(0, 100))
lines(xx, dbetapr(xx, 5, 3, 2), col = "red")
hist(pbetapr(x, 5, 3, 2))
plot(ecdf(x), xlim = c(0, 100))
lines(xx, pbetapr(xx, 5, 3, 2), col = "red")