View source for Laplace distribution

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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|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 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

$$\beta+X_1-X_2,$$

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:

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

and

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

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top"> P.S. Laplace, "Théorie analytique des probabilités", Paris (1812)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> W. Feller, [[Feller, "An introduction to probability theory and its  applications"|"An introduction to probability theory and its  applications"]], '''2''', Wiley (1971)</TD></TR></table>

====Comments====


====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  E. Lukacs,   "Characteristic functions" , Griffin  (1970)</TD></TR></table>