dist.Pareto | R Documentation |
These functions provide the density, distribution function, quantile function, and random generation for the pareto distribution.
dpareto(x, alpha, log=FALSE)
ppareto(q, alpha)
qpareto(p, alpha)
rpareto(n, alpha)
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 |
log |
Logical. If |
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.
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
.
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"))
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