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Crofton's Formula

Let n points xi_1, ..., xi_n be randomly distributed on a domain S, and let H be some event that depends on the positions of the n points. Let S^' be a domain slightly smaller than S but contained within it, and let deltaS be the part of S not in S^'. Let P[H] be the probability of event H, s be the measure of S, and deltaS the measure of deltaS, then Crofton's formula states that

deltaP[H]=n(P[Hxi_1 in deltaS]-P[H])s^(-1)deltas

(Solomon 1978, p. 99).

See also

Crofton's Integrals

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References

Pfiefer, R. E. "The Historical Development of J. J. Sylvester's Four Point Problem." Math. Mag. 62, 309-317, 1989.Ruben, H. and Reed, W. J. "A More General Form of the Theory of Crofton." J. Appl. Prob. 10, 479-482, 1973.Solomon, H. "Crofton's Theorem and Sylvester's Problem in Two and Three Dimensions." Ch. 5 in Geometric Probability. Philadelphia, PA: SIAM, pp. 97-125, 1978.

Referenced on Wolfram|Alpha

Crofton's Formula

Cite this as:

Weisstein, Eric W. "Crofton's Formula." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/CroftonsFormula.html

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