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 \(X\sim \operatorname{Laplace}(\mu,b)\), 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 \(X\sim \operatorname{Laplace}(\mu,b)\!\), then \(kX+b\sim \operatorname{Laplace}(k\mu+b,kb)\!\)
• If \(X \sim \operatorname{Laplace}(0,b)\!\), then \(|X| \sim \operatorname{Exponential}(\tfrac{1}{b})\!\) (Exponential distribution)
• If \(X \sim \operatorname{Exponential}(\tfrac{1}{\lambda})\!\) and \(Y \sim \operatorname{Exponential}(\tfrac{1}{\lambda})\!\), then \(X-Y \sim \operatorname{Laplace}\left(0,\lambda\right)\!\)
• If \(X \sim \operatorname{Laplace}(\mu,\tfrac{1}{b})\!\), then \(|X-\mu| \sim \operatorname{Exponential}(b)\!\)
• If \(X_i \sim \operatorname{Normal}(0,1)\!\) for \(i={1,2,3,4}\!\) then \(X_1 X_2 - X_3 X_4 \sim \operatorname{Laplace}(0,1)\!\) (Normal distribution)
• If \(X_i \sim \operatorname{Normal}(0,1)\!\) for \(i={1,2,3,4}\!\), then \(X_1 X_2 + X_3 X_4 \sim \operatorname{Laplace}(0,1)\!\)
• If \(X_i \sim \operatorname{Laplace}(\mu,b)\!\), then \(\frac{2}{b \sum_{i=1}^n |X_i-\mu|} \sim \chi^2(2n) \! \) (Chi-square distribution)
• If \(X \sim \operatorname{Laplace}(\mu,b)\) and \(Y \sim \operatorname{Laplace}(\mu,b)\) then \( \tfrac{|X-\mu|}{|Y-\mu|} \sim \operatorname{F}(2,2) \) (F-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

Example

Suppose that the return of a certain stock has a Laplace distribution with \(\mu=5\) and \(b=2\). Compute the probability that the stock will have a return between 6 and 10.

We can compute this as follows:

\[P(6 \le X\le 10)=\sum_{x=6}^{10}\frac{1}{2\times 2}\exp(-\frac{|x-5|}{2})=0.262223\]

The figure below shows this result using SOCR distributions