Laplace Distribution

$\begin{figure}\begin{center}\BoxedEPSF{LaplaceDistribution.epsf scaled 650}\end{center}\end{figure}$

Also called the Double Exponential Distribution. It is the distribution of differences between two independent variates with identical Exponential Distributions (Abramowitz and Stegun 1972, p. 930).

$\displaystyle P(x)$	$\textstyle =$	$\displaystyle {1\over 2b}e^{-\vert x-\mu\vert/b}$	(1)
$\displaystyle D(x)$	$\textstyle =$	$\displaystyle {\textstyle{1\over 2}}[1+\mathop{\rm sgn}\nolimits (x-\mu)(1-e^{-\vert x-\mu\vert/b})].$	(2)

The Moments about the Mean $\mu_n$ are related to the Moments about 0 by

$\begin{displaymath} \mu_n=\sum_{j=0}^n{n\choose j}(-1)^{n-j}\mu'_j\mu^{n-j}, \end{displaymath}$

(3)

where ${n\choose k}$ is a Binomial Coefficient, so

$\displaystyle \mu_n$	$\textstyle =$	$\displaystyle \sum_{j=0}^n\sum_{k=0}^{\left\lfloor{j/2}\right\rfloor } (-1)^{n-j}{n\choose j}{j\choose 2k}b^{2k}\mu^{n-2k}\Gamma(2k+1)$
	$\textstyle =$	$\displaystyle \left\{\begin{array}{ll} n! b^n & \mbox{for $n$\ even}\\ 0 & \mbox{for $n$\ odd,}\end{array}\right.$	(4)

where $\left\lfloor{x}\right\rfloor$ is the Floor Function and $\Gamma(2k+1)$ is the Gamma Function.

The Moments can also be computed using the Characteristic Function,

$\begin{displaymath} \phi(t)\equiv\int_{-\infty}^\infty e^{itx} P(x)\,dx = {1\over 2b} \int_{-\infty}^\infty e^{itx}e^{-\vert x-\mu\vert/b}\,dx. \end{displaymath}$

(5)

Using the Fourier Transform of the Exponential Function

$\begin{displaymath} {\mathcal F}[e^{-2\pi k_0\vert x\vert}] = {1\over\pi} {k_0\over k^2+{k_0}^2} \end{displaymath}$

(6)

gives

$\begin{displaymath} \phi(t)={e^{i\mu t}\over 2b} {{2\over b}\over t^2+\left({1\over b}\right)^2} = {e^{i\mu t}\over 1+b^2t^2}. \end{displaymath}$

(7)

The Moments are therefore

$\begin{displaymath} \mu_n=(-i)^n\phi(0)=(-i)^n\left[{d^n\phi\over dt^n}\right]_{t=0}. \end{displaymath}$

(8)

The Mean, Variance, Skewness, and Kurtosis are

$\displaystyle \mu$	$\textstyle =$	$\displaystyle \mu$	(9)
$\displaystyle \sigma^2$	$\textstyle =$	$\displaystyle 2b^2$	(10)
$\displaystyle \gamma_1$	$\textstyle =$	$\displaystyle 0$	(11)
$\displaystyle \gamma_2$	$\textstyle =$	$\displaystyle 3.$	(12)

References

Abramowitz, M. and Stegun, C. A. (Eds.). Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, 1972.