Laplace Distribution — Definition, Formula & Examples

The Laplace distribution is a continuous probability distribution that looks like two exponential distributions placed back-to-back around a center point. It has heavier tails than the normal distribution, meaning extreme values are more likely.

A continuous random variable $X$ follows a Laplace distribution with location parameter $\mu \in \mathbb{R}$ and scale parameter $b > 0$ , written $X \sim \text{Laplace}(\mu, b)$ , if its probability density function is $f(x) = \frac{1}{2b}\exp\!\left(-\frac{|x - \mu|}{b}\right)$ for all $x \in \mathbb{R}$ . Its mean is $\mu$ and its variance is $2b^2$ .

Key Formula

f(x) = \frac{1}{2b}\,\exp\!\left(-\frac{|x - \mu|}{b}\right)

Where:

$x$ = Value of the random variable
$\mu$ = Location parameter (mean and median)
$b$ = Scale parameter (b > 0); standard deviation is b√2

How It Works

The Laplace distribution is symmetric about

\mu

, and the scale parameter

b

controls how spread out it is. Smaller

b

produces a sharper peak; larger

b

flattens it. Because the density decays exponentially (rather than as a Gaussian), the tails are heavier. You can evaluate probabilities by integrating the PDF, or use the CDF directly: for

x \geq \mu

F(x) = 1 - \frac{1}{2}\exp\!\left(-\frac{x-\mu}{b}\right)

Worked Example

Problem: A random variable follows a Laplace distribution with μ = 0 and b = 2. Find the probability density at x = 3.

Substitute into the PDF: Plug μ = 0, b = 2, and x = 3 into the formula.

f(3) = \frac{1}{2(2)}\,\exp\!\left(-\frac{|3 - 0|}{2}\right) = \frac{1}{4}\,e^{-1.5}

Evaluate: Compute the exponential and multiply.

e^{-1.5} \approx 0.2231 \quad\Rightarrow\quad f(3) \approx \frac{0.2231}{4} = 0.0558

Answer: The probability density at x = 3 is approximately 0.0558.

Visualization

Why It Matters

The Laplace distribution is the foundation of L1 regularization (Lasso) in machine learning, where a Laplace prior on model weights encourages sparsity. It also appears in robust statistics and signal processing when data contain more outliers than a normal model would predict.

Common Mistakes

Mistake: Confusing the Laplace scale parameter b with the standard deviation.

Correction: The variance is 2b², so the standard deviation is b√2, not b. Always convert if a problem asks for the standard deviation.