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. It is a special case of the
generalized normal distribution and the
hyperbolic distribution. Continuous symmetric distributions that have exponential tails, like the Laplace distribution, but which have probability density functions that are differentiable at the mode include the
logistic distribution,
hyperbolic secant distribution, and the
Champernowne distribution.
If X has a Laplace distribution, then Y = eX has a log-Laplace distribution; conversely, if X has a log-Laplace distribution, then its
logarithm has a Laplace distribution.
Probability of a Laplace being greater than another
Let be independent laplace random variables: and , and we want to compute .
The probability of can be reduced (using the properties below) to , where . This probability is equal to
When , both expressions are replaced by their limit as :
To compute the case for , note that
since when
Relation to the exponential distribution
A Laplace random variable can be represented as the difference of two
independent and identically distributed (
iid) exponential random variables.[2] 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
Sargan distributions
Sargan distributions are a system of distributions of which the Laplace distribution is a core member. A th order Sargan distribution has density[3][4]
for parameters . The Laplace distribution results for .
Statistical inference
Given independent and identically distributed samples , the
maximum likelihood (MLE) estimator of is the sample
median,[5]
The Laplacian distribution has been used in speech recognition to model priors on
DFT coefficients [6] and in JPEG image compression to model AC coefficients [7] generated by a
DCT.
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
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.
The Laplace distribution has applications in finance. For example, S.G. Kou developed a model for financial instrument prices incorporating a Laplace distribution (in some cases an
asymmetric Laplace distribution) to address problems of
skewness,
kurtosis and the
volatility smile that often occur when using a normal distribution for pricing these instruments.[10][11]
The Laplace distribution, being a composite or double distribution, is applicable in situations where the lower values originate under different external conditions than the higher ones so that they follow a different pattern.[12]
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.
A variate can also be generated as the difference of two
i.i.d. random variables. Equivalently, can also be generated as the
logarithm of the ratio of two
i.i.d. uniform random variables.
History
This distribution is often referred to as "Laplace's first law of errors". He published it in 1774, modeling the frequency of an error as an exponential function of its magnitude once its sign was disregarded. Laplace would later replace this model with his "second law of errors", based on the normal distribution, after the discovery of the
central limit theorem.[13][14]
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.[15]
^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, Edwin Bidwell (1923). "First and Second Laws of Error". Journal of the American Statistical Association. 18 (143). Informa UK Limited: 841–851.
doi:
10.1080/01621459.1923.10502116.
ISSN0162-1459. This article incorporates text from this source, which is in the
public domain.