       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

```

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