extraDistr (version 1.8.1)

Pareto: Pareto distribution

Description

Density, distribution function, quantile function and random generation for the Pareto distribution.

Usage

dpareto(x, a = 1, b = 1, log = FALSE)
ppareto(q, a = 1, b = 1, lower.tail = TRUE, log.p = FALSE)
qpareto(p, a = 1, b = 1, lower.tail = TRUE, log.p = FALSE)
rpareto(n, a = 1, b = 1)

Arguments

x, q

vector of quantiles.

a, b

positive valued scale and location parameters.

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

Probability density function $$f(x)=\frac{a{b}^a}{x^{a+1}}$$

Cumulative distribution function $$F(x)=1-{\left(\frac{b}{x}\right)}^a$$

Quantile function $$F^{-1}(p)=\frac{b}{(1-p{)}^{1-a}}$$

References

Krishnamoorthy, K. (2006). Handbook of Statistical Distributions with Applications. Chapman & Hall/CRC

Examples

x <- rpareto(1e5, 5, 16)
xx <- seq(-100, 100, by = 0.001)
hist(x, 100, freq = FALSE)
lines(xx, dpareto(xx, 5, 16), col = "red")
hist(ppareto(x, 5, 16))
plot(ecdf(x))
lines(xx, ppareto(xx, 5, 16), col = "red", lwd = 2)