Laplace Distribution

The Laplace distribution, also called the double exponential distribution, is the distribution of differences between two independent variates with identical exponential distributions (Abramowitz and Stegun 1972, p. 930). It had probability density function and cumulative distribution functions given by

			(1)
			(2)

It is implemented in the Wolfram Language as LaplaceDistribution[mu, beta].

The moments about the mean  are related to the moments about 0 by

(3)

where  is a binomial coefficient, so

			(4)
			(5)

where  is the floor function and  is the gamma function.

The moments can also be computed using the characteristic function,

(6)

Using the Fourier transform of the exponential function

(7)

gives

(8)

(Abramowitz and Stegun 1972, p. 930). The moments are therefore

(9)

The mean, variance, skewness, and kurtosis excess are

			(10)
			(11)
			(12)
			(13)

REFERENCES:

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

Papoulis, A. Probability, Random Variables, and Stochastic Processes, 2nd ed. New York: McGraw-Hill, p. 104, 1984.