# AP Statistics Curriculum 2007 Laplace

## Contents

### 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(μ,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)
• μ 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)
• μ 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 μ = 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