Description Usage Arguments Details Value Note Author(s) References See Also Examples

Density, distribution function, quantile function, and random generation
for the Pareto distribution with parameters `location`

and `shape`

.

1 2 3 4 |

`x` |
vector of quantiles. |

`q` |
vector of quantiles. |

`p` |
vector of probabilities between 0 and 1. |

`n` |
sample size. If |

`location` |
vector of (positive) location parameters. |

`shape` |
vector of (positive) shape parameters. The default is |

Let *X* be a Pareto random variable with parameters `location=`

*η*
and `shape=`

*θ*. The density function of *X* is given by:

*f(x; η, θ) = \frac{θ η^θ}{x^{θ + 1}}, \; η > 0, \; θ > 0, \; x ≥ η*

The cumulative distribution function of *X* is given by:

*F(x; η, θ) = 1 - (\frac{η}{x})^θ*

and the *p*'th quantile of *X* is given by:

*x_p = η (1 - p)^{-1/θ}, \; 0 ≤ p ≤ 1*

The mode, mean, median, variance, and coefficient of variation of *X* are given by:

*Mode(X) = η*

*E(X) = \frac{θ η}{θ - 1}, \; θ > 1*

*Median(X) = x_{0.5} = 2^{1/θ} η*

*Var(X) = \frac{θ η^2}{(θ - 1)^2 (θ - 1)}, \; θ > 2*

*CV(X) = [θ (θ - 2)]^{-1/2}, \; θ > 2*

`dpareto`

gives the density, `ppareto`

gives the distribution function,
`qpareto`

gives the quantile function, and `rpareto`

generates random
deviates.

The Pareto distribution is named after Vilfredo Pareto (1848-1923), a professor
of economics. It is derived from Pareto's law, which states that the number of
persons *N* having income *≥ x* is given by:

*N = A x^{-θ}*

where *θ* denotes Pareto's constant and is the shape parameter for the
probability distribution.

The Pareto distribution takes values on the positive real line. All values must be
larger than the “location” parameter *η*, which is really a threshold
parameter. There are three kinds of Pareto distributions. The one described here
is the Pareto distribution of the first kind. Stable Pareto distributions have
*0 < θ < 2*. Note that the *r*'th moment only exists if
*r < θ*.

The Pareto distribution is related to the
exponential distribution and
logistic distribution as follows.
Let *X* denote a Pareto random variable with `location=`

*η* and
`shape=`

*θ*. Then *log(X/η)* has an exponential distribution
with parameter `rate=`

*θ*, and *-log\{ [(X/η)^θ] - 1 \}*
has a logistic distribution with parameters `location=`

*0* and
`scale=`

*1*.

The Pareto distribution has a very long right-hand tail. It is often applied in the study of socioeconomic data, including the distribution of income, firm size, population, and stock price fluctuations.

Steven P. Millard (EnvStats@ProbStatInfo.com)

Forbes, C., M. Evans, N. Hastings, and B. Peacock. (2011). Statistical Distributions. Fourth Edition. John Wiley and Sons, Hoboken, NJ.

Johnson, N. L., S. Kotz, and N. Balakrishnan. (1994).
*Continuous Univariate Distributions, Volume 1*.
Second Edition. John Wiley and Sons, New York.

`epareto`

, `eqpareto`

, Exponential,
Probability Distributions and Random Numbers.

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 | ```
# Density of a Pareto distribution with parameters location=1 and shape=1,
# evaluated at 2, 3 and 4:
dpareto(2:4, 1, 1)
#[1] 0.2500000 0.1111111 0.0625000
#----------
# The cdf of a Pareto distribution with parameters location=2 and shape=1,
# evaluated at 3, 4, and 5:
ppareto(3:5, 2, 1)
#[1] 0.3333333 0.5000000 0.6000000
#----------
# The 25'th percentile of a Pareto distribution with parameters
# location=1 and shape=1:
qpareto(0.25, 1, 1)
#[1] 1.333333
#----------
# A random sample of 4 numbers from a Pareto distribution with parameters
# location=3 and shape=2.
# (Note: the call to set.seed simply allows you to reproduce this example.)
set.seed(10)
rpareto(4, 3, 2)
#[1] 4.274728 3.603148 3.962862 5.415322
``` |

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