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A continuous probability distribution with density
 
A continuous probability distribution with density
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057460/l0574601.png" /></td> </tr></table>
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$$p(x)=\frac12\alpha e^{-\alpha|x-\beta|},\quad-\infty<x<\infty,$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057460/l0574602.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057460/l0574603.png" />, is a shift parameter and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057460/l0574604.png" /> is a scale parameter. The density of the Laplace distribution is symmetric about the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057460/l0574605.png" />, and the derivative of the density has a discontinuity at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057460/l0574606.png" />. The characteristic function of the Laplace distribution with parameters <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057460/l0574607.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057460/l0574608.png" /> is
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where $\beta$, $-\infty<\beta<\infty$, is a shift parameter and $\alpha>0$ is a scale parameter. The density of the Laplace distribution is symmetric about the point $x=\beta$, and the derivative of the density has a discontinuity at $x=\beta$. The characteristic function of the Laplace distribution with parameters $\alpha$ and $\beta$ is
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057460/l0574609.png" /></td> </tr></table>
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$$e^{it\beta}\frac{1}{1+t^2/\alpha^2}.$$
  
The Laplace distribution has finite moments of any order. In particular, its [[Mathematical expectation|mathematical expectation]] is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057460/l05746010.png" /> and its variance (cf. [[Dispersion|Dispersion]]) is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057460/l05746011.png" />.
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The Laplace distribution has finite moments of any order. In particular, its [[Mathematical expectation|mathematical expectation]] is $\beta$ and its variance (cf. [[Dispersion|Dispersion]]) is $2/\alpha^2$.
  
The Laplace distribution was first introduced by P. Laplace [[#References|[1]]] and is often called the  "first law of Laplacefirst law of Laplace" , in contrast to the  "second law of Laplacesecond law of Laplace" , as the [[Normal distribution|normal distribution]] is sometimes called. The Laplace distribution is also called the two-sided exponential distribution, on account of the fact that the Laplace distribution coincides with the distribution of the random variable
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The Laplace distribution was first introduced by P. Laplace [[#References|[1]]] and is often called the  "first law of Laplace", in contrast to the  "second law of Laplace", as the [[Normal distribution|normal distribution]] is sometimes called. The Laplace distribution is also called the two-sided exponential distribution, on account of the fact that the Laplace distribution coincides with the distribution of the random variable
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057460/l05746012.png" /></td> </tr></table>
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$$\beta+X_1-X_2,$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057460/l05746013.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057460/l05746014.png" /> are independent random variables that have the same [[Exponential distribution|exponential distribution]] with density <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057460/l05746015.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057460/l05746016.png" />. The Laplace distribution with density <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057460/l05746017.png" /> and the [[Cauchy distribution|Cauchy distribution]] with density <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057460/l05746018.png" /> are related in the following way:
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where $X_1$ and $X_2$ are independent random variables that have the same [[Exponential distribution|exponential distribution]] with density $\alpha e^{-\alpha x}$, $x>0$. The Laplace distribution with density $e^{-|x|}/2$ and the [[Cauchy distribution|Cauchy distribution]] with density $1/(\pi(1+x^2))$ are related in the following way:
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057460/l05746019.png" /></td> </tr></table>
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$$\frac12\int\limits_{-\infty}^\infty e^{itx}e^{-|x|}dx=\frac{1}{1+t^2}$$
  
 
and
 
and
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057460/l05746020.png" /></td> </tr></table>
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$$\frac1\pi\int\limits_{-\infty}^\infty e^{-itx}\frac{1}{1+t^2}dt=e^{-|x|}.$$
  
 
====References====
 
====References====

Latest revision as of 07:53, 20 August 2014

A continuous probability distribution with density

$$p(x)=\frac12\alpha e^{-\alpha|x-\beta|},\quad-\infty<x<\infty,$$

where $\beta$, $-\infty<\beta<\infty$, is a shift parameter and $\alpha>0$ is a scale parameter. The density of the Laplace distribution is symmetric about the point $x=\beta$, and the derivative of the density has a discontinuity at $x=\beta$. The characteristic function of the Laplace distribution with parameters $\alpha$ and $\beta$ is

$$e^{it\beta}\frac{1}{1+t^2/\alpha^2}.$$

The Laplace distribution has finite moments of any order. In particular, its mathematical expectation is $\beta$ and its variance (cf. Dispersion) is $2/\alpha^2$.

The Laplace distribution was first introduced by P. Laplace [1] and is often called the "first law of Laplace", in contrast to the "second law of Laplace", as the normal distribution is sometimes called. The Laplace distribution is also called the two-sided exponential distribution, on account of the fact that the Laplace distribution coincides with the distribution of the random variable

$$\beta+X_1-X_2,$$

where $X_1$ and $X_2$ are independent random variables that have the same exponential distribution with density $\alpha e^{-\alpha x}$, $x>0$. The Laplace distribution with density $e^{-|x|}/2$ and the Cauchy distribution with density $1/(\pi(1+x^2))$ are related in the following way:

$$\frac12\int\limits_{-\infty}^\infty e^{itx}e^{-|x|}dx=\frac{1}{1+t^2}$$

and

$$\frac1\pi\int\limits_{-\infty}^\infty e^{-itx}\frac{1}{1+t^2}dt=e^{-|x|}.$$

References

[1] P.S. Laplace, "Théorie analytique des probabilités", Paris (1812)
[2] W. Feller, "An introduction to probability theory and its applications", 2, Wiley (1971)

Comments

References

[a1] E. Lukacs, "Characteristic functions" , Griffin (1970)
How to Cite This Entry:
Laplace distribution. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Laplace_distribution&oldid=33035
This article was adapted from an original article by A.V. Prokhorov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article