# AP Statistics Curriculum 2007 Laplace

### From Socr

(→Related Distributions) |
(→Applications) |
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*Insurance claims | *Insurance claims | ||

*Structural changes in switching-regime model and Kalman filter | *Structural changes in switching-regime model and Kalman filter | ||

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+ | ===Example=== | ||

+ | Suppose that the return of a certain stock has a Laplace distribution with <font size=3><math>\mu=5</math></font> and <font size=3><math>b=2</math></font>. Compute the probability that the stock will have a return between 6 and 10. | ||

+ | |||

+ | We can compute this as follows: | ||

+ | |||

+ | :<math>P(6 \le X\le 10)=\sum_{x=6}^{10}\frac{1}{2\times 2}\exp(-\frac{|x-5|}{2})=0.262223</math> | ||

+ | |||

+ | The figure below shows this result using [http://socr.ucla.edu/htmls/dist/Laplace_Distribution.html SOCR distributions] | ||

+ | <center>[[Image:Laplace.jpg|600px]]</center> |

## Revision as of 21:05, 11 July 2011

## 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

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

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

**Expectation**:

**Variance**: The gamma variance is

### Related Distributions

- If , then
- If , then (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:

The figure below shows this result using SOCR distributions