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
Support
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)
E(X) = , Var(X) =
Note that the pdf of the Laplace distribution is symmetric about the location parameter μ.
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.