# densities: The Laplace Distribution In tsxtreme: Bayesian Modelling of Extremal Dependence in Time Series

## Description

Density, distribution function, quantile function and random generation for the Laplace distribution with location parameter `loc` and scale parameter `scale`.

## Usage

 ```1 2 3 4``` ```dlapl(x, loc = 0, scale = 1, log = FALSE) plapl(q, loc = 0, scale = 1, lower.tail = TRUE, log.p = FALSE) qlapl(p, loc = 0, scale = 1, lower.tail = TRUE, log.p = FALSE) rlapl(n, loc = 0, scale = 1) ```

## Arguments

 `x,q` vector of quantiles. `p` vector of probabilities. `n` number of samples to generate. `loc` vector of location parameters. `scale` vector of scale parameters. These must be positive. `lower.tail` logical; if TRUE (default), probabilities are Pr(X≤ x), otherwise Pr(X>x). `log,log.p` logical; if TRUE, probabilities p are given as log(p).

## Details

If `loc` or `scale` are not specified, they assume the default values of 0 and 1 respectively.

The Laplace distribution has density

f(x) = \exp(- |x-μ|/σ)/(2σ)

where μ is the location parameter and σ is the scale parameter.

## Value

`dlapl` gives the density, `plapl` gives the distribution function, `qlapl` gives the quantile function, and `rlapl` generates random deviates.

The length of the result is determined by `n` in `rlapl`, and is the maximum of the lengths of the numerical arguments for the other functions. Standard `R` vector operations are to be assumed.

If `sd`=0, the limit as `sd` decreases to 0 is returned, i.e., a point mass at `loc`. The case `sd`<0 is an error and nothing is returned.

## Warning

Some checks are done previous to standard evaluation, but vector computations have not yet been tested thoroughly! Typically vectors not having lengths multiple of each other return an error.

 ``` 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15``` ```## evaluate the density function on a grid of values x <- seq(from=-5, to=5, by=0.1) fx <- dlapl(x, loc=1, scale=.5) ## generate random samples of a mixture of Laplace distributions rnd <- rlapl(1000, loc=c(-5,-3,2), scale=0.5) ## an alternative: rnd <- runif(1000) rnd <- qlapl(rnd, loc=c(-5,-3,2), scale=0.5) ## integrate the Laplace density on [a,b] a <- -1 b <- 7 integral <- plapl(b)-plapl(a) ```