Difference between revisions of "Laplace distribution"
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Revision as of 09:15, 4 May 2012
A continuous probability distribution with density
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where ,
, is a shift parameter and
is a scale parameter. The density of the Laplace distribution is symmetric about the point
, and the derivative of the density has a discontinuity at
. The characteristic function of the Laplace distribution with parameters
and
is
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The Laplace distribution has finite moments of any order. In particular, its mathematical expectation is and its variance (cf. Dispersion) is
.
The Laplace distribution was first introduced by P. Laplace [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 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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where and
are independent random variables that have the same exponential distribution with density
,
. The Laplace distribution with density
and the Cauchy distribution with density
are related in the following way:
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and
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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) |
Laplace distribution. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Laplace_distribution&oldid=14706