The Laplace distribution

Source: R/dist_laplace.R

[Stable]

The Laplace distribution, also known as the double exponential distribution, is a continuous probability distribution that is symmetric around its location parameter.

dist_laplace(mu, sigma)

Arguments

mu

The location parameter (mean) of the Laplace distribution.

sigma

The positive scale parameter of the Laplace distribution.

Details

We recommend reading this documentation on pkgdown which renders math nicely. https://pkg.mitchelloharawild.com/distributional/reference/dist_laplace.html

In the following, let \(X\) be a Laplace random variable with location parameter mu = \(\mu\) and scale parameter sigma = \(\sigma\).

Support: \(R\), the set of all real numbers

Mean: \(\mu\)

Variance: \(2\sigma^2\)

Probability density function (p.d.f):

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

Cumulative distribution function (c.d.f):

$$ F(x) = \begin{cases} \frac{1}{2} \exp\left(\frac{x - \mu}{\sigma}\right) & \text{if } x < \mu \\ 1 - \frac{1}{2} \exp\left(-\frac{x - \mu}{\sigma}\right) & \text{if } x \geq \mu \end{cases} $$

Moment generating function (m.g.f):

$$ E(e^{tX}) = \frac{\exp(\mu t)}{1 - \sigma^2 t^2} \text{ for } |t| < \frac{1}{\sigma} $$

See also

extraDistr::Laplace

Examples

dist <- dist_laplace(mu = c(0, 2, -1), sigma = c(1, 2, 0.5))

dist
#> <distribution[3]>
#> [1] Laplace(0, 1)    Laplace(2, 2)    Laplace(-1, 0.5)
mean(dist)
#> [1]  0  2 -1
variance(dist)
#> [1] 2.0 8.0 0.5
skewness(dist)
#> [1] 0 0 0
kurtosis(dist)
#> [1] 3 3 3

generate(dist, 10)
#> [[1]]
#>  [1]  1.36711211  0.41731557  0.16381514 -0.66221683 -1.76133439  1.15992222
#>  [7] -1.63859856 -0.08484774 -3.42374992  0.29025085
#> 
#> [[2]]
#>  [1]  1.84052207  1.25746756  3.43749205 -1.72211123  1.09198626 -0.01766638
#>  [7]  7.10521169  3.79418775  8.30334282 -1.32393260
#> 
#> [[3]]
#>  [1] -0.79896108 -0.37234528 -1.28012352 -0.94928966 -1.45270781  0.08059396
#>  [7] -0.81478629  0.09223017 -1.98185496 -1.96915562
#> 

density(dist, 0)
#> [1] 0.50000000 0.09196986 0.13533528
density(dist, 0, log = TRUE)
#> [1] -0.6931472 -2.3862944 -2.0000000

cdf(dist, 1)
#> [1] 0.8160603 0.3032653 0.9908422

quantile(dist, 0.7)
#> [1]  0.5108256  3.0216512 -0.7445872