AP Statistics Curriculum 2007 Laplace
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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