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 along the abscissa, 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. Definitions Probability density function A random variable has a distribution if its probability density function is Here, is a location parameter and , which is sometimes referred to as the "diversity", is a scale parameter. If and , the positive half-line is exactly an exponential distribution scaled by 1/2. 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. Cumulative distribution function The Laplace distribution is easy to integrate (if one distinguishes two symmetric cases) due to the use of the absolute value function. Its cumulative distribution function is as follows: The inverse cumulative distribution function is given by Properties Moments Related distributions If then . If then . If then (exponential distribution). If then ． If then . If then (exponential power distribution). If (normal distribution) then and . 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 . Note that if , then with , which in turn equals the exponential distribution . Given an integer , if (gamma distribution, using characterization), then (infinite divisibility) If X has a Laplace distribution, then Y = eX has a log-Laplace distribution; conversely, if X has a log-Laplace distribution, then its