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Map-Airy Distribution

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MapAiryDistribution

The map-Airy distribution is a statistical distribution having probability density function and distribution function

P(x)=2e^(-2x^3/3)[xAi(x^2)-Ai^'(x^2)]
(1)
D(x)=1/3-2x^5(_2F_2(7/6,5/3;7/3,8/3;-4/3x^3))/(15·3^(2/3)Gamma(5/3))-x^4(_2F_2(5/6,4/3;5/3,7/3;-4/3x^3))/(6·3^(1/3)Gamma(4/3))+x^2(_2F_2(1/6,2/3;1/3,5/3;-4/3x^3))/(3^(2/3)Gamma(2/3))+2x(_2F_2(-1/6,1/3;-1/3,4/3;-4/3x^3))/(3^(1/3)Gamma(1/3)),
(2)

where Ai(x) is the Airy function and Ai^'(x) is its derivative. The density is normalized with

int_(-infty)^inftyP(x)dx=1.
(3)

The mean is 0 and

int_(-infty)^0xP(x)dx=(Gamma(1/3))/(2pi·3^(1/6))
(4)
int_0^inftyxP(x)dx=-(Gamma(1/3))/(2pi·3^(1/6)),
(5)

but the second moment mu_2 is undefined.

See also

Airy Functions

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References

Banderier, C.; Flajolet, P.; Schaeffer, G.; and Soria, M. "Planar Maps and Airy Phenomena." In Automata, Languages and Programming. Proceedings of the 27th International Colloquium (ICALP 2000) held at the University of Geneva, Geneva, July 9-15, 2000 (Ed. U. Montanari, J. D. P. Rolim, and E. Welzl). Berlin: Springer-Verlag, pp. 388-402, 2000.Banderier, C.; Flajolet, P.; Schaeffer, G.; and Soria, M. "Random Maps, Coalescing Saddles, Singularity Analysis, and Airy Phenomena." Random Structures Alg. 19, 194-246, 2001. http://www-lipn.univ-paris13.fr/~banderier/Papers/rsa.ps.

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Map-Airy Distribution

Cite this as:

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

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