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The unit circle {(x, y) :
The unit circle {(x, y) : x2+y2 = 1} is divided into three arcs by choosing three random points A,B,C on the circle (independently and uniformly), forming arcs between A and B, between A and C, and between B and C. Let L be the length of the arc containing the point (1, 0). What is E(L)? Study this by working through the following steps.
(a) Explain what is wrong with the following argument: “The total length of the arcs is 2π , the circumference of the circle. So by symmetry and linearity, each arc has length 2π/3 on average. Referring to the arc containing (1, 0) is just a way to specify one of the arcs (it wouldn’t matter if (1, 0) were replaced by (0,−1) or any other specific point on the circle in the statement of the problem). So the expected value of L is 2π/3.”
(b) Let the arc containing (1, 0) be divided into two pieces: the piece extending counterclockwise from (1, 0) and the piece extending clockwise from (1, 0).
Write L = L1 + L2, where L1 and L2 are the lengths of the counterclockwise and clockwise pieces, respectively. Find the CDF, PDF, and expected value of L1.
(c) Use (b) to find E(L).
