Probability Playground: The Laplace Distribution

The Laplace (or double exponential) distribution has the form of two exponential distributions joined back-to-back around a location parameter μ. It arises naturally as the difference between two independent and identically distributed exponential random variables.

Parameter	Range	Description
μ	−∞ < μ < ∞	Location parameter
β	β > 0	Scale parameter

Probability Density Function

f (x; μ, β) = \frac{1}{2 β} e^{- | x - μ | / β}

F (x; μ, β) = {\begin{array}{cl} \frac{1}{2} e^{\frac{x - μ}{β}} & if & x \leq μ \\ 1 - \frac{1}{2} e^{\frac{- (x - μ)}{β}} & if & x > μ \end{array}

Support

- \infty < x < \infty

Mean

Variance

Example	μ	β
Two people fishing both wait on average 2 hours to catch a fish. Let X be the difference between the waiting times for the first and second person.	0.000	2.000
In an 8-bit grayscale image, the difference in brightness between successive pixels can be approximately modeled as a Laplace(0, 8) random variable.	0.000	8.000
Workers at a call center wait 10 seconds on average for the next call. Let X be the difference between the waiting times for two consecutive calls.	0.0000	10.00

X ∼ Laplace(μ, β)

μ = β =

E(X) = , Var(X) =

Note that the pdf of the Laplace distribution is symmetric about the location parameter μ.

Since the pdf of the Laplace distribution is symmetric about the location parameter μ, the cdf has the property that F(μ - x) = 1 - F(μ + x).

The illustration above shows two values X₁ and X₂ chosen independently and at random from an exponential(β) distribution. The random variable X₁ − X₂ + μ has a Laplace(μ, β) distribution, where μ denotes a location parameter.

The simulation above shows two values X₁ and X₂ chosen independently and at random from an exponential(β) distribution. The light blue line shows the value of X₁ − X₂ + μ, where μ denotes a location parameter. The distribution of this value has a Laplace(μ, β) distribution. The histogram accumulates the results of each simulation.

Y = |X − μ| ∼ Exponential(β)

E(Y) = , Var(Y) =

Proof

X ∼ Laplace(μ, β)

Y = |X − μ| ∼ Exponential(β)