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Laplace Distribution

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LaplaceDistribution

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

P(x)=1/(2b)e^(-|x-mu|/b)
(1)
D(x)=1/2[1+sgn(x-mu)(1-e^(-|x-mu|/b))].
(2)

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

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

mu_n=sum_(j=0)^n(n; j)(-1)^(n-j)mu_j^'mu^(n-j),
(3)

where (n; k) is a binomial coefficient, so

mu_n=sum_(j=0)^(n)sum_(k=0)^(|_j/2_|)(-1)^(n-j)(n; j)(j; 2k)b^(2k)mu^(n-2k)Gamma(2k+1)
(4)
={n!b^n for n even; 0 for n odd,
(5)

where |_x_| is the floor function and Gamma(2k+1) is the gamma function.

The moments can also be computed using the characteristic function,

phi(t)=int_(-infty)^inftye^(itx)P(x)dx=1/(2b)int_(-infty)^inftye^(itx)e^(-|x-mu|/b)dx.
(6)

Using the Fourier transform of the exponential function

F_x[e^(-2pik_0|x|)](k)=1/pi(k_0)/(k^2+k_0^2)
(7)

gives

phi(t)=(e^(imut))/(1+b^2t^2)
(8)

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

mu_n=(-i)^nphi(0)=(-i)^n[(d^nphi)/(dt^n)]_(t=0).
(9)

The mean, variance, skewness, and kurtosis excess are

mu=mu
(10)
sigma^2=2b^2
(11)
gamma_1=0
(12)
gamma_2=3.
(13)

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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.

Referenced on Wolfram|Alpha

Laplace Distribution

Cite this as:

Weisstein, Eric W. "Laplace Distribution." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/LaplaceDistribution.html

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