题目
Consider a circle of radius with an inscribed equilateral triangle. A chord is drawn 'at random.' What is the probability that the chord is longer than the side of the triangle? Compute the answer under each of the following three methods of selecting a random chord:
(a) Random endpoints: Fix one endpoint on the circle and choose the other endpoint uniformly on the circumference. (b) Random midpoint: Choose the midpoint of the chord uniformly in the disk. (c) Random radius: Choose a radius, then choose a point uniformly along that radius as the midpoint of the chord.
For each method, set up and evaluate the relevant integral or geometric argument.
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a
b
c
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In quantitative finance, model risk arises when different but seemingly equivalent parameterizations yield different prices. Give a concrete example where a 'natural' choice of probability measure on a financial model leads to ambiguity analogous to Bertrand's paradox, and explain how practitioners resolve this.
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