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
