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. 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 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, The MLE estimator of is the mean absolute deviation from the median, revealing a link between the Laplace distribution and least absolute deviations. A correction for small samples can be applied as follows: (see: exponential distribution#Parameter estimation). Occurrence and applications The Laplacian distribution has been used in speech recognition to model priors on DFT coefficients and in JPEG image compression to model AC coefficients 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 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 for the coefficients. In hydrology the Laplace distribution is applied to extreme events such as annual maximum one-day rainfalls and river discharges. The blue picture, made