AP Statistics Curriculum 2007 Laplace

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(Laplace Distribution)
(Related Distributions)
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===Related Distributions===
===Related Distributions===
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*If <math>X\sim Laplace(\mu,b)\!</math>, then <math>kX+b\sim Laplace(k\mu+b,kb)\!</math>
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*If <math>X\sim \operatorname{Laplace}(\mu,b)\!</math>, then <math>kX+b\sim \operatorname{Laplace}(k\mu+b,kb)\!</math>
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*If <math>X \sim Laplace(0,b)\!</math>, then <math>|X| \sim Exponential(\tfrac{1}{b})\!</math> ([[exponential distribution]])
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*If <math>X \sim \operatorname{Laplace}(0,b)\!</math>, then <math>|X| \sim \operatorname{Exponential}(\tfrac{1}{b})\!</math> ([[Exponential distribution]])
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*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>
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*If <math>X \sim \operatorname{Laplace}(\mu,\tfrac{1}{b})\!</math>, then <math>|X-\mu| \sim \operatorname{Exponential}(b)\!</math>
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* 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]])
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* 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>
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* 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]])
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*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 21:14, 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

\frac{1}{2b}\exp(-\frac{|x-\mu|}{b})

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


\left\{\begin{matrix}
\frac{1}{2}\exp(\frac{x-\mu}{b}) & \mbox{if }x < \mu
\\[8pt]
1-\frac{1}{2}\exp(-\frac{x-\mu}{b}) & \mbox{if }x \geq \mu
\end{matrix}\right.

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

M(t)=\frac{\exp(\mu t)}{1-b^2 t^2} \mbox{ for }|t|<\frac{1}{b}


Expectation:

E(X)=\mu\!


Variance: The gamma variance is

Var(X)=2b^2\!

Related Distributions

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:

P(6 \le X\le 10)=\sum_{x=6}^{10}\frac{1}{2\times 2}\exp(-\frac{|x-5|}{2})=0.262223

The figure below shows this result using SOCR distributions

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