ReIns (version 1.0.10)

Pareto: The Pareto distribution

Description

Density, distribution function, quantile function and random generation for the Pareto distribution (type I).

Usage

dpareto(x, shape, scale = 1, log = FALSE)
ppareto(x, shape, scale = 1, lower.tail = TRUE, log.p = FALSE)
qpareto(p, shape, scale = 1, lower.tail = TRUE, log.p = FALSE)
rpareto(n, shape, scale = 1)

Value

dpareto gives the density function evaluated in \(x\), ppareto the CDF evaluated in \(x\) and qpareto the quantile function evaluated in \(p\). The length of the result is equal to the length of \(x\) or \(p\).

rpareto returns a random sample of length \(n\).

Arguments

x

Vector of quantiles.

p

Vector of probabilities.

n

Number of observations.

shape

The shape parameter of the Pareto distribution, a strictly positive number.

scale

The scale parameter of the Pareto distribution, a strictly positive number. Its default value is 1.

log

Logical indicating if the densities are given as \(\log(f)\), default is FALSE.

lower.tail

Logical indicating if the probabilities are of the form \(P(X\le x)\) (TRUE) or \(P(X>x)\) (FALSE). Default is TRUE.

log.p

Logical indicating if the probabilities are given as \(\log(p)\), default is FALSE.

Author

Tom Reynkens.

Details

The Cumulative Distribution Function (CDF) of the Pareto distribution is equal to \(F(x) = 1-(x/scale)^{-shape}\) for all \(x \ge scale\) and \(F(x)=0\) otherwise. Both shape and scale need to be strictly positive.

See Also

tPareto, GPD, Distributions

Examples

# Plot of the PDF
x <- seq(1, 10, 0.01)
plot(x, dpareto(x, shape=2), xlab="x", ylab="PDF", type="l")

# Plot of the CDF
x <- seq(1, 10, 0.01)
plot(x, ppareto(x, shape=2), xlab="x", ylab="CDF", type="l")

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