Beta prime distribution

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

Usage

dbetaprime(x, shape1, shape2, log = FALSE)

pbetaprime(q, shape1, shape2, lower.tail = TRUE, log.p = FALSE)

qbetaprime(p, shape1, shape2, lower.tail = TRUE, log.p = FALSE)

rbetaprime(n, shape1, shape2)

Arguments

x, q

vector of quantiles

shape1, shape2

non-negative shape parameters of the corresponding Beta distribution

log, log.p

logical; if TRUE, probabilities/ densities \(p\) are returned 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 random values to return.

Value

dbetaprime gives the density, pbetaprime gives the distribution function, qbetaprime gives the quantile function, and rbetaprime generates random deviates.

Details

This implementation allows for automatic differentiation with RTMB.

If \(X \sim \text{Beta}(\alpha, \beta)\), then \(\frac{X}{1-X} \sim \text{Betaprime}(\alpha, \beta)\)

Examples

x <- rbetaprime(1, 2, 1)
d <- dbetaprime(x, 2, 1)
p <- pbetaprime(x, 2, 1)
q <- qbetaprime(p, 2, 1)