# AP Statistics Curriculum 2007 Laplace

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*If <math>X\sim Laplace(\mu,b)\!</math>, then <math>kX+b\sim Laplace(k\mu+b,kb)\!</math> | *If <math>X\sim Laplace(\mu,b)\!</math>, then <math>kX+b\sim Laplace(k\mu+b,kb)\!</math> | ||

*If <math>X \sim Laplace(0,b)\!</math>, then <math>|X| \sim Exponential(\tfrac{1}{b})\!</math> ([[exponential distribution]]) | *If <math>X \sim Laplace(0,b)\!</math>, then <math>|X| \sim Exponential(\tfrac{1}{b})\!</math> ([[exponential distribution]]) | ||

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===Applications=== | ===Applications=== |

## Revision as of 20:59, 11 July 2011

### 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