# BetaPrime: Beta prime distribution In extraDistr: Additional Univariate and Multivariate Distributions

## Description

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

## Usage

 ```1 2 3 4 5 6 7``` ```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 ≤ 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(α, β), then X/(1-X) ~ BetaPrime(α, β).

Probability density function

f(x) = ((x/σ)^(α-1) * (1 + x/σ)^(-α-β)) / (B(α,β) * σ)

Cumulative distribution function

F(x) = pbeta((x/σ)/(1+(x/σ)), α, β)

`Beta`
 ```1 2 3 4 5 6``` ```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) ```