# Laplace: Laplace distribution

## Description

Density, distribution function, quantile function and random generation for the Laplace distribution.

## Usage

dlaplace(x, mu = 0, sigma = 1, log = FALSE)plaplace(q, mu = 0, sigma = 1, lower.tail = TRUE, log.p = FALSE)qlaplace(p, mu = 0, sigma = 1, lower.tail = TRUE, log.p = FALSE)rlaplace(n, mu = 0, sigma = 1)

## Arguments

x, q

vector of quantiles.

mu, sigma

location and scale parameters. Scale must be positive.

log, log.p

logical; if TRUE, probabilities 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.

n

number of observations. If length(n) > 1, the length is taken to be the number required.

## Details

Probability density function $$f(x) = \frac{1}{2\sigma} \exp\left(-\left|\frac{x-\mu}{\sigma}\right|\right)$$

Cumulative distribution function $$F(x) = \left\{\begin{array}{ll} \frac{1}{2} \exp\left(\frac{x-\mu}{\sigma}\right) & x < \mu \\ 1 - \frac{1}{2} \exp\left(\frac{x-\mu}{\sigma}\right) & x \geq \mu \end{array}\right.$$

Quantile function $$F^{-1}(p) = \left\{\begin{array}{ll} \mu + \sigma \log(2p) & p < 0.5 \\ \mu - \sigma \log(2(1-p)) & p \geq 0.5 \end{array}\right.$$

## References

Krishnamoorthy, K. (2006). Handbook of Statistical Distributions with Applications. Chapman & Hall/CRC

Forbes, C., Evans, M. Hastings, N., & Peacock, B. (2011). Statistical Distributions. John Wiley & Sons.

## Examples

# NOT RUN {
x <- rlaplace(1e5, 5, 16)
hist(x, 100, freq = FALSE)
curve(dlaplace(x, 5, 16), -200, 200, n = 500, col = "red", add = TRUE)
hist(plaplace(x, 5, 16))
plot(ecdf(x))
curve(plaplace(x, 5, 16), -200, 200, n = 500, col = "red", lwd = 2, add = TRUE)

# }