AP Statistics Curriculum 2007 Laplace

Laplace Distribution

Definition: Laplace distribution is a distribution that is symmetrical and more “peaky” than a normal distribution. The dispersion of the data around the mean is higher than that of a normal distribution. Laplace distribution is also sometimes called the double exponential distribution.

Probability density function: For X~Laplace(\(\mu\),b), the Laplace probability density function is given by

\[\frac{1}{2b}exp(-\frac{|x-\mu|}{b})\]

where

• e is the natural number (e = 2.71828…)
• b is a scale parameter (determines the profile of the distribution)
• \(\mu\) is the mean
• x is a random variable

Cumulative density function: The Laplace cumulative distribution function is given by

\[ \left\{\begin{matrix} \frac{1}{2}\exp(\frac{x-\mu}{b}) & \mbox{if }x < \mu \\[8pt] 1-\frac{1}{2}\exp(-\frac{x-\mu}{b}) & \mbox{if }x \geq \mu \end{matrix}\right. \]

where

• e is the natural number (e = 2.71828…)
• b is a scale parameter (determines the profile of the distribution)
• \(\mu\) is the mean
• x is a random variable

Moment generating function: The Laplace moment-generating function is

\[M(t)=\frac{\exp(\mu t)}{1-b^2 t^2} \mbox{ for }|t|<\frac{1}{b}\]

Expectation:

\[E(X)=\mu\!\]

Variance: The gamma variance is

\[Var(X)=2b^2\!\]

Related Distributions

• If \(X\sim Laplace(\mu,b)\!\), then \(kX+b\sim Laplace(k\mu+b,kb)\!\)
• If \(X \sim Laplace(0,b)\!\), then \(|X| \sim Exponential(\tfrac{1}{b})\!\) (exponential distribution)

Applications

The Laplace distribution is used for modeling in signal processing, various biological processes, finance, and economics. Examples of events that may be modeled by Laplace distribution include:

• Credit risk and exotic options in financial engineering
• Insurance claims
• Structural changes in switching-regime model and Kalman filter