The Laplace distribution is a continuous probability distribution with probability density function given by:

subject to the following conditions:

The Laplace variate with location parameter 0 and scale parameter b is related to two independent Exponential variates $\mathrm{E1}$ and $\mathrm{E2}$ by Laplace(0,b) ~ E1 - E2.

Note that the Laplace command is inert and should be used in combination with the RandomVariable command.

$\mathrm{with}\left(\mathrm{Statistics}\right)\:$

$X≔\mathrm{RandomVariable}\left(\mathrm{Laplace}\left(a\,b\right)\right)\:$

$\mathrm{PDF}\left(X\,u\right)$

$\frac{{\ⅇ}^{-\frac{\left|-u+a\right|}{b}}}{2b}$

$\mathrm{PDF}\left(X\,0.5\right)$

$\frac{0.5000000000{\ⅇ}^{-\frac{1.\left|-0.5+a\right|}{b}}}{b}$

$\mathrm{Mean}\left(X\right)$

$a$

$\mathrm{Variance}\left(X\right)$

$2b^2$

Evans, Merran; Hastings, Nicholas; and Peacock, Brian. Statistical Distributions 3rd ed. Hoboken: Wiley, 2000.

Johnson, Norman L.; Kotz, Samuel; and Balakrishnan, N. Continuous Univariate Distributions. 2nd ed. 2 vols. Hoboken: Wiley, 1995.

Stuart, Alan, and Ord, Keith. Kendall's Advanced Theory of Statistics. 6th ed. London: Edward Arnold, 1998. Vol. 1: Distribution Theory.