# AP Statistics Curriculum 2007 Laplace

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'''Probability density function''': For X~Laplace($\mu$,b), the Laplace probability density function is given by
'''Probability density function''': For X~Laplace($\mu$,b), the Laplace probability density function is given by - :$\frac{1}{2b}exp(-\frac{|x-\mu|}{b})$ + :$\frac{1}{2b}\exp(-\frac{|x-\mu|}{b})$ where where

## 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 $X\sim Laplace(\mu,b)\!$, 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

### 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