dist.Pareto {LaplacesDemon}

Pareto Distribution

Description

These functions provide the density, distribution function, quantile function, and random generation for the pareto distribution.

Usage

dpareto(x, alpha, log=FALSE)
ppareto(q, alpha)
qpareto(p, alpha)
rpareto(n, alpha)

Arguments

x, q

These are each a vector of quantiles.

p

This is a vector of probabilities.

n

This is the number of observations, which must be a positive integer that has length 1.

alpha

This is the shape parameter \alpha, which must be positive.

log

Logical. If log=TRUE, then the logarithm of the density or result is returned.

Details

• Application: Continuous Univariate

• Density: p(\theta) = \frac{\alpha}{\theta^{\alpha+1}}, \theta \ge 1

• Inventor: Vilfredo Pareto (1848-1923)

• Notation 1: \theta \sim \mathcal{PA}(\alpha)

• Notation 2: p(\theta) = \mathcal{PA}(\theta | \alpha)

• Parameter 1: shape parameter \alpha > 0

• Mean: E(\theta) = \frac{\alpha}{\alpha - 1}

• Variance: var(\theta) = \frac{\alpha}{(\alpha-1)^2(\alpha-2)}, \alpha > 2

• Mode: mode(\theta) = 1

The Pareto distribution, sometimes called the Bradford distribution, is related to the exponential distribution. The gamma distribution is the conjugate prior distribution for the shape parameter \alpha in the Pareto distribution. The Pareto distribution is the conjugate prior distribution for the range parameters of a uniform distribution. An extension, elsewhere, is the symmetric Pareto distribution.

Value

dpareto gives the density, ppareto gives the distribution function, qpareto gives the quantile function, and rpareto generates random deviates.

See Also

dexp, dlnorm, dlnormp, dnorm, dnormp, dnormv.

Examples

library(LaplacesDemon)
x <- dpareto(1,1)
x <- ppareto(0.5,1)
x <- qpareto(0.5,1)
x <- rpareto(10,1)

#Plot Probability Functions
x <- seq(from=1, to=5, by=0.01)
plot(x, dpareto(x,0.1), ylim=c(0,1), type="l", main="Probability Function",
     ylab="density", col="red")
lines(x, dpareto(x,0.5), type="l", col="green")
lines(x, dpareto(x,1), type="l", col="blue")
legend(2, 0.9, expression(alpha==0.1, alpha==0.5, alpha==1),
     lty=c(1,1,1), col=c("red","green","blue"))