# AP Statistics Curriculum 2007 Laplace

### From Socr

(→Laplace Distribution) |
(→Laplace Distribution) |
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*e is the natural number (e = 2.71828…) | *e is the natural number (e = 2.71828…) | ||

*b is a scale parameter (determines the profile of the distribution) | *b is a scale parameter (determines the profile of the distribution) | ||

- | *<math>\mu</math> is the mean | + | *<font size="3"><math>\mu</math></font> is the mean |

*x is a random variable | *x is a random variable | ||

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*e is the natural number (e = 2.71828…) | *e is the natural number (e = 2.71828…) | ||

*b is a scale parameter (determines the profile of the distribution) | *b is a scale parameter (determines the profile of the distribution) | ||

- | *<math>\mu</math> is the mean | + | *<font size="3"><math>\mu</math></font> is the mean |

*x is a random variable | *x is a random variable | ||

## Revision as of 21:00, 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