dist.Pareto: Pareto Distribution In LaplacesDemonR/LaplacesDemonCpp: C++ Extension for LaplacesDemon

Description

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

Usage

 ```1 2 3 4``` ```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) = alpha / theta^(alpha + 1), theta >= 1

• Inventor: Vilfredo Pareto (1848-1923)

• Notation 1: theta ~ PA(alpha)

• Notation 2: p(theta) = PA(theta | alpha)

• Parameter 1: shape parameter alpha > 0

• Mean: E(theta) = alpha / (alpha - 1)

• Variance: var(theta) = 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.

`dexp`, `dlnorm`, `dlnormp`, `dnorm`, `dnormp`, `dnormv`.
 ``` 1 2 3 4 5 6 7 8 9 10 11 12 13 14``` ```library(LaplacesDemonCpp) 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")) ```