Laplace distribution

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)

References

[a1]	E. Lukacs, "Characteristic functions" , Griffin (1970)