📂Probability Distribution

Laplace Distribution

Definition¹

For $\mu \in \mathbb{R}$ and $b > 0$, a continuous probability distribution $\operatorname{Laplace}(\mu, b)$ with the following probability density function is called the Laplace distribution.

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

Explanation

Relationship with the Normal Distribution

Although it looks similar to the normal distribution, it has an absolute value $| x - \mu |$ instead of a square, giving it a sharper shape. Near $0$, the Laplace distribution decreases much more rapidly, while farther from $0$, the normal distribution decreases more rapidly.

Relationship with the Exponential Distribution

Particularly when $\mu = 0$ and $b = 1$ in the Laplace distribution, it coincides with the exponential distribution $\frac{1}{2} \exp(1)$ when $x \ge 0$. It can be seen as an extension of the exponential distribution defined in $x \ge 0$ to $x \in \mathbb{R}$. If $X \sim \operatorname{Laplace}(0, b)$, then $|X| \sim \exp(b^{-1})$.

Basic Properties

Moment Generating Function

The moment generating function for the Laplace distribution is given by:

$$m(t) = \dfrac{1}{1 - t^{2}b^{2}}\exp(\mu t) \qquad \text{for } |t| < \dfrac{1}{b}$$

Mean and Variance

If $X \sim \operatorname{Laplace}(\mu, b)$, then

$$E(X) = \mu$$ $$\Var(X) = 2b^{2}$$

Maximum Likelihood Estimator

Suppose that a random sample $\mathbf{X} := \left( X_{1} , \cdots , X_{n} \right) \sim \operatorname{Laplace}(\mu, b)$ is given.

The maximum likelihood estimator for $(\mu, b)$ is as follows:

$$\hat{\mu} = \text{median}(\mathbf{x}_{1}, \cdots, \mathbf{x}_{n})$$

$$\hat{b} = \dfrac{1}{n} \sum\limits_{k=1}^{n} |x_{k} - \mu|$$

Visualization

The following is a Julia code snippet illustrating the probability density function of the Laplace distribution as a GIF.

@time using LaTeXStrings
@time using Distributions
@time using Plots

cd(@__DIR__)

x = -10:0.1:10
B = collect(0.1:0.1:5.0); append!(B, reverse(B))

animation = @animate for b ∈ B
    plot(x, pdf.(Laplace(0, b), x),
     color = :black,
     label = "b = $(round(b, digits = 2))", size = (400,300))
    xlims!(-10,10); ylims!(0,1); title!(L"\mathrm{pdf\,of\,} \operatorname{Laplace}(0, b)")
end
gif(animation, "pdf.gif", fps = 15)

plot(x, pdf.(Laplace(0, 1), x), lw=2, label="Laplace(0, 1)", color=:royalblue, dpi=200)
plot!(x, pdf.(Normal(0, 1), x), lw=2, label="Normal(0, 1)", color=:tomato)