In probability theory and statistics, the Laplace distribution is a continuous probability distribution named after Pierre-Simon Laplace. It is also sometimes called the double exponential distribution, because it can be thought of as two exponential distributions (with an additional location parameter) spliced together back-to-back, although the term is also sometimes used to refer to the Gumbel distribution. The difference between two independent identically distributed exponential random variables is governed by a Laplace distribution, as is a Brownian motion evaluated at an exponentially distributed random time. Increments of Laplace motion or a variance gamma process evaluated over the time scale also have a Laplace distribution.
Probability density function
Cumulative distribution function
Probability density function
The probability density function of the Laplace distribution is also reminiscent of the normal distribution; however, whereas the normal distribution is expressed in terms of the squared difference from the mean , the Laplace density is expressed in terms of the absolute difference from the mean. Consequently, the Laplace distribution has fatter tails than the normal distribution.
where is the generalized exponential integral function .
- If then .
- If then . (Exponential distribution)
- If then ．
- If then .
- If then . (Exponential power distribution)
- If (Normal distribution) then .
- If then . (Chi-squared distribution)
- If then . (F-distribution)
- If (Uniform distribution) then .
- If and (Bernoulli distribution) independent of , then .
- If and independent of , then ．
- If has a Rademacher distribution and then .
- If and independent of , then .
- If (geometric stable distribution) then .
- The Laplace distribution is a limiting case of the hyperbolic distribution.
- If with (Rayleigh distribution) then .
- Given an integer , if (gamma distribution, using characterization), then (infinite divisibility)
Relation to the exponential distribution
A Laplace random variable can be represented as the difference of two iid exponential random variables. One way to show this is by using the characteristic function approach. For any set of independent continuous random variables, for any linear combination of those variables, its characteristic function (which uniquely determines the distribution) can be acquired by multiplying the corresponding characteristic functions.
Consider two i.i.d random variables . The characteristic functions for are
respectively. On multiplying these characteristic functions (equivalent to the characteristic function of the sum of the random variables ), the result is
This is the same as the characteristic function for , which is
Estimation of parameters
Given independent and identically distributed samples , the maximum likelihood estimator of is the sample median, and the maximum likelihood estimator of is the Mean Absolute Deviation from the Median
(revealing a link between the Laplace distribution and least absolute deviations).
Occurrence and applications
- The addition of noise drawn from a Laplacian distribution, with scaling parameter appropriate to a function's sensitivity, to the output of a statistical database query is the most common means to provide differential privacy in statistical databases.
- In regression analysis, the least absolute deviations estimate arises as the maximum likelihood estimate if the errors have a Laplace distribution.
- The Lasso can be thought of as a Bayesian regression with a Laplacian prior.
- In hydrology the Laplace distribution is applied to extreme events such as annual maximum one-day rainfalls and river discharges. The blue picture, made with CumFreq, illustrates an example of fitting the Laplace distribution to ranked annually maximum one-day rainfalls showing also the 90% confidence belt based on the binomial distribution. The rainfall data are represented by plotting positions as part of the cumulative frequency analysis.
Generating values from the Laplace distribution
Given a random variable drawn from the uniform distribution in the interval , the random variable
has a Laplace distribution with parameters and . This follows from the inverse cumulative distribution function given above.
This distribution is often referred to as Laplace's first law of errors. He published it in 1774 when he noted that the frequency of an error could be expressed as an exponential function of its magnitude once its sign was disregarded.
Keynes published a paper in 1911 based on his earlier thesis wherein he showed that the Laplace distribution minimised the absolute deviation from the median.
- Kotz, Samuel; Kozubowski, Tomasz J.; Podgórski, Krzysztof (2001). The Laplace distribution and generalizations: a revisit with applications to Communications, Economics, Engineering and Finance. Birkhauser. pp. 23 (Proposition 2.2.2, Equation 2.2.8). ISBN 9780817641665.
- Everitt, B.S. (2002) The Cambridge Dictionary of Statistics, CUP. ISBN 0-521-81099-X
- Johnson, N.L., Kotz S., Balakrishnan, N. (1994) Continuous Univariate Distributions, Wiley. ISBN 0-471-58495-9. p. 60
- Robert M. Norton (May 1984). "The Double Exponential Distribution: Using Calculus to Find a Maximum Likelihood Estimator". The American Statistician. American Statistical Association. 38 (2): 135–136. doi:10.2307/2683252. JSTOR 2683252.
- Eltoft, T.; Taesu Kim; Te-Won Lee (2006). "On the multivariate Laplace distribution" (PDF). IEEE Signal Processing Letters. 13 (5): 300–303. doi:10.1109/LSP.2006.870353.
- Minguillon, J.; Pujol, J. (2001). "JPEG standard uniform quantization error modeling with applications to sequential and progressive operation modes" (PDF). Journal of Electronic Imaging. 10 (2): 475–485. doi:10.1117/1.1344592.
- CumFreq for probability distribution fitting
- A collection of composite distributions
- Laplace, P-S. (1774). Mémoire sur la probabilité des causes par les évènements. Mémoires de l’Academie Royale des Sciences Presentés par Divers Savan, 6, 621–656
- Wilson EB (1923) First and second laws of error. JASA 18, 143
- Keynes JM (1911) The principal averages and the laws of error which lead to them. J Roy Stat Soc, 74, 322–331