Laplace Distribution
المؤلف:
Abramowitz, M. and Stegun, I. A.
المصدر:
Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, 1972.
الجزء والصفحة:
...
7-4-2021
1871
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
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
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
=1/pi(k_0)/(k^2+k_0^2)](https://mathworld.wolfram.com/images/equations/LaplaceDistribution/NumberedEquation3.gif) |
(7)
|
gives
 |
(8)
|
(Abramowitz and Stegun 1972, p. 930). The moments are therefore
![mu_n=(-i)^nphi(0)=(-i)^n[(d^nphi)/(dt^n)]_(t=0).](https://mathworld.wolfram.com/images/equations/LaplaceDistribution/NumberedEquation5.gif) |
(9)
|
The mean, variance, skewness, and kurtosis excess are
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.
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