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圆弧覆盖圆

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Circle covering by arcs

概率 P(a,n) 为 n 个角尺寸为 a 的随机弧完全覆盖圆的圆周（对于单位圆周长的圆）是

P(a,n)=sum_(k=0)^(|_1/a_|)(-1)^k(n; k)(1-ka)^(n-1),

其中 |_x_| 是 向下取整函数 (Solomon 1978, p. 75)。这最初由 Stevens (1939) 正确给出，尽管 Whitworth (1897)、Baticle (1935)、Garwood (1940)、Darling (1953) 和 Shepp (1972) 获得了部分结果。

概率为 n 个弧留下正好 l 个间隙由下式给出

P_(l gaps)(a,n)=(n; l)sum_(j=l)^(|_1/a_|)(-1)^(j-l)(n-l; j-l)(1-ja)^(n-1)

(Stevens 1939; Solomon 1978, pp. 75-76)。

另请参阅

圆点选取, 圆线选取

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参考文献

Baticle, M. "Le problème des répartitions." Comptes Rendus Acad. Sci. Paris 201, 862-864, 1935.Fisher, R. A. "Tests of Significance in Harmonic Analysis." Proc. Roy. Soc. London Ser. A 125, 54-59, 1929.Fisher, R. A. "On the Similarity of the Distributions Found for the Test of Significance in Harmonic Analysis, and in Stevens's Problem in Geometric Probability." Eugenics 10, 14-17, 1940.Darling, D. A. "On a Class of Problems Related to the Random Division of an Interval." Ann. Math. Stat. 24, 239-253, 1953.Garwood, F. "An Application to the Theory of Probability of the Operation of Vehicular-Controlled Traffic Signals." J. Roy. Stat. Soc. Suppl. 7, 65-77, 1940.Shepp, L. A. "Covering the Circle with Random Arcs." Israel J. Math. 11, 328-345, 1972.Siegel, A. F. Random Coverage Problems in Geometric Probability with an Application to Time Series Analysis. Ph.D. thesis. Stanford, CA: Stanford University, 1977.Solomon, H. "Covering a Circle Circumference and a Sphere Surface." Ch. 4 in Geometric Probability. Philadelphia, PA: SIAM, pp. 75-96, 1978.Stevens, W. L. "Solution to a Geometrical Problem in Probability." Ann. Eugenics 9, 315-320, 1939.Whitworth, W. A. DCC Exercises in Choice and Chance. 1897. Reprinted New York: Hafner, 1965.

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圆弧覆盖圆

请引用为

Weisstein, Eric W. "圆弧覆盖圆。" 来自 MathWorld--一个 Wolfram 网络资源。 https://mathworld.net.cn/CircleCoveringbyArcs.html

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