AP Statistics Curriculum 2007 Laplace
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===Related Distributions=== | ===Related Distributions=== | ||
- | *If <math>X\sim Laplace(\mu,b)\!</math>, then <math>kX+b\sim Laplace(k\mu+b,kb)\!</math> | + | *If <math>X\sim \operatorname{Laplace}(\mu,b)\!</math>, then <math>kX+b\sim \operatorname{Laplace}(k\mu+b,kb)\!</math> |
- | *If <math>X \sim Laplace(0,b)\!</math>, then <math>|X| \sim Exponential(\tfrac{1}{b})\!</math> ([[ | + | *If <math>X \sim \operatorname{Laplace}(0,b)\!</math>, then <math>|X| \sim \operatorname{Exponential}(\tfrac{1}{b})\!</math> ([[Exponential distribution]]) |
+ | *If <math>X \sim \operatorname{Exponential}(\tfrac{1}{\lambda})\!</math> and <math>Y \sim \operatorname{Exponential}(\tfrac{1}{\lambda})\!</math>, then <math>X-Y \sim \operatorname{Laplace}\left(0,\lambda\right)\!</math> | ||
+ | *If <math>X \sim \operatorname{Laplace}(\mu,\tfrac{1}{b})\!</math>, then <math>|X-\mu| \sim \operatorname{Exponential}(b)\!</math> | ||
+ | * If <math>X_i \sim \operatorname{Normal}(0,1)\!</math> for <math>i={1,2,3,4}\!</math> then <math>X_1 X_2 - X_3 X_4 \sim \operatorname{Laplace}(0,1)\!</math> ([[Normal distribution]]) | ||
+ | * If <math>X_i \sim \operatorname{Normal}(0,1)\!</math> for <math>i={1,2,3,4}\!</math>, then <math>X_1 X_2 + X_3 X_4 \sim \operatorname{Laplace}(0,1)\!</math> | ||
+ | * If <math>X_i \sim \operatorname{Laplace}(\mu,b)\!</math>, then <math>\frac{2}{b \sum_{i=1}^n |X_i-\mu|} \sim \chi^2(2n) \! </math> ([[Chi-square distribution]]) | ||
+ | *If <math>X \sim \operatorname{Laplace}(\mu,b)</math> and <math>Y \sim \operatorname{Laplace}(\mu,b)</math> then <math> \tfrac{|X-\mu|}{|Y-\mu|} \sim \operatorname{F}(2,2) </math> ([[F-distribution]]) | ||
===Applications=== | ===Applications=== |
Revision as of 21:16, 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)
- 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