# AP Statistics Curriculum 2007 Laplace

### From Socr

## 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 , 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)
- If and , then
- If , then
- If for then (Normal distribution)
- If for , then
- If , then (Chi-square distribution)
- If and then (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:

The figure below shows this result using SOCR distributions