distributions

*To*: mathgroup at yoda.physics.unc.edu*Subject*: distributions*From*: deb at alexandria.lcs.mit.edu (David E. Burmaster)*Date*: Thu, 23 Dec 93 15:52:49 -0500

Dear MathGroup The Levy family of probability distributions is defined as Integrate[Exp[ -t^a] Cos[t x], {t, 0, Infinity}]/Pi where a is parameter and x is the random variable. In general, there is no closed-form for this integral. For the case a = 1, the integral becomes the Cauchy Distribution. For the case a = 2, the integral becomes a Gaussian distribution. A colleague and I are interested in cases for 1 2 a 2 2 as a possible model for distributions with long tails to the right, i.e., large positive skew. For the case a = (3/2), Mma gives an analytical expression for the integral that includes many Gamma functions and HypergeometricPFQ functions. So, now for the question. We want to Plot the PDF for x from x = 0 to x = 8 or so. At x = 0, the PDF is undefined (it should be 0???) but the real problem is apparent numeric instability for 0 < x < 0.04 or so. The problem is that the graph produced by Plot in Mma appears to oscillate from -10^30 to +10^30 -- clearly a problem. Question: Can any of you propose a method to contain these wild (and incorrect) oscillations near the origin??? Many thanks for you help Dave Dave David E. Burmaster, Ph.D. Alceon Corporation Cambridge, MA 02238