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. 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. Random variate generation 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, 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. See also Generalized normal distribution#Symmetric version Multivariate Laplace distribution Besov measure, a generalisation of the Laplace distribution to function spaces Cauchy distribution, also called the "Lorentzian distribution" (the Fourier transform of the Laplace) Characteristic function (probability theory) References External links Continuous distributions Compound probability distributions Pierre-Simon Laplace Exponential family distributions Location-scale family probability distributions Geometric stable distributions Infinitely divisible probability distributions