These functions provide information about the Pareto distribution.
`dpareto`

gives the density, `ppareto`

gives the distribution
function, `qpareto`

gives the quantile function and `rpareto`

generates random
deviates.

## Usage

```
rpareto(n = 1L, shape = 0, scale = 1)
dpareto(x, shape = 1, scale = 1, log = FALSE)
ppareto(q, shape = 1, scale = 1, lower.tail = TRUE, log.p = FALSE)
qpareto(p, shape = 1, scale = 1, lower.tail = TRUE, log.p = FALSE)
```

## Arguments

- n
integer number of observations.

- shape
shape parameter (must be positive).

- scale
scale parameter (must be positive).

- x, q
vector of quantiles.

- log, log.p
logical; if

`TRUE`

, probabilities/densities`p`

are given as`log(p)`

.- lower.tail
logical; if

`TRUE`

(default), probabilities are \(P(X \le x)\), otherwise \(P(X > x)\).- p
vector of probabilities.

## Value

`rpareto`

generates random deviates.

`dpareto`

gives the density.

`ppareto`

gives the distribution function.

`qpareto`

gives the quantile function.

## Details

If `shape`

or `scale`

are not specified, they assume the default values of `1`

.

The Pareto distribution with scale \(\theta\) and shape \(\xi\) has density

$$f(x) = \xi \theta^\xi / (x + \theta)^(\xi + 1)$$

The support is \(x \ge 0\).

The Expected value exists if \(\xi > 1\) and is equal to

$$E(X) = \theta / (\xi - 1)$$

k-th moments exist in general for \(k < \xi\).

## References

https://en.wikipedia.org/wiki/Pareto_distribution - named Lomax therein.

## Examples

```
x <- rpareto(1000, shape = 10, scale = 5)
xx <- seq(-1, 10, 0.01)
hist(x, breaks = 100, freq = FALSE, xlim = c(-1, 10))
lines(xx, dpareto(xx, shape = 10, scale = 5))
plot(xx, dpareto(xx, shape = 10, scale = 5), type = "l")
lines(xx, dpareto(xx, shape = 3, scale = 5), col = "red", lwd = 2)
plot(xx, dpareto(xx, shape = 10, scale = 10), type = "l")
lines(xx, dpareto(xx, shape = 10, scale = 5), col = "blue", lwd = 2)
lines(xx, dpareto(xx, shape = 10, scale = 20), col = "red", lwd = 2)
```