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Lévy Distribution


 F_k[P_N(k)](x)=F_k[exp(-N|k|^beta)](x),

where F is the Fourier transform of the probability P_N(k) for N-step addition of random variables. Lévy showed that beta in (0,2) for P(x) to be nonnegative. The Lévy distribution has infinite variance and sometimes infinite mean. The case beta=1 gives a Cauchy distribution, while beta=2 gives a normal distribution.

The Lévy distribution is implemented in the Wolfram Language as LevyDistribution[mu, sigma].


See also

Cauchy Distribution, Lévy Flight, Normal Distribution

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Cite this as:

Weisstein, Eric W. "Lévy Distribution." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/LevyDistribution.html

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