exponential random variable with mean 1; for compute: for compute: where This algorithm yields a random variable . For a detailed proof see. Given the formulas for simulation of a standard stable random variable, we can easily simulate a stable random variable for all admissible values of the parameters , , and using the following property. If then is . For (and ) the CMS method reduces to the well known Box-Muller transform for generating Gaussian random variables. Many other approaches have been proposed in the literature, including application of Bergström and LePage series expansions, see and, respectively. However, the CMS method is regarded as the fastest and the most accurate. Applications Stable distributions owe their importance in both theory and practice to the generalization of the central limit theorem to random variables without second (and possibly first) order moments and the accompanying self-similarity of the stable family. It was the seeming departure from normality along with the demand for a self-similar model for financial data (i.e. the shape of the distribution for yearly asset price changes should resemble that of the constituent daily or monthly price changes) that led Benoît Mandelbrot to propose that cotton prices follow an alpha-stable distribution with equal to 1.7. Lévy distributions are frequently found in analysis of critical behavior and financial data. They are also found in spectroscopy as a general expression for a quasistatically pressure broadened spectral line. The Lévy distribution of solar flare waiting time events (time between flare events) was demonstrated for CGRO BATSE hard x-ray solar flares in December 2001. Analysis of the Lévy statistical signature revealed that two different memory signatures were evident; one related to the solar cycle and the second whose origin appears to be associated with a localized or combination of localized solar active region effects. Other analytic cases A number of cases of analytically expressible stable distributions are known. Let the stable distribution be expressed by then we know: The Cauchy Distribution is given by The Lévy distribution is given by The Normal distribution is given by Let be a Lommel function, then: Let and denote the Fresnel Integrals then: Let be the modified Bessel function of the second kind then: If the denote the hypergeometric functions then: with the latter being the Holtsmark distribution. Let be a Whittaker function, then: See also Lévy flight Lévy process Other "power law" distributions Pareto distribution Zeta distribution