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圓弧覆蓋圓

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Circle covering by arcs

機率 P(a,n) 為 n 個角尺寸為 a 的隨機弧完全覆蓋圓的圓周（對於單位圓周長的圓）是

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

其中 |_x_| 是 向下取整函式 (Solomon 1978, p. 75)。這最初由 Stevens (1939) 正確給出，儘管 Whitworth (1897)、Baticle (1935)、Garwood (1940)、Darling (1953) 和 Shepp (1972) 獲得了部分結果。

機率為 n 個弧留下正好 l 個間隙由下式給出

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)。

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參考文獻

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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圓弧覆蓋圓

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Weisstein, Eric W. "圓弧覆蓋圓。" 來自 --一個 Wolfram 網路資源。 https://mathworld.tw/CircleCoveringbyArcs.html

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