bv_cvxbook_extra_exercises

# M give the optimal worstcase mean return achieved by

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Unformatted text preview: . . . , xn be a set of nonnegative numbers with positive sum, which typically represent the wealth or income of n individuals in some group. The Lorentz curve is a plot of the fraction fi of total wealth held by the i poorest individuals, i T x (j ) , fi = (1/1 x) i = 0, . . . , n, j =1 versus i/n, where x(j ) denotes the j th smallest of the numbers {x1 , . . . , xn }, and we take f0 = 0. The Lorentz curve starts at (0, 0) and ends at (1, 1). Interpreted as a continuous curve (as, say, n → ∞) the Lorentz curve is convex and increasing, and lies on or below the straight line joining the endpoints. The curve coincides with this straight line, i.e., fi = (i/n), if and only if the wealth is distributed equally, i.e., the xi are all equal. The Gini coeﬃcient is deﬁned as twice the area between the straight line corresponding to uniform wealth distribution and the Lorentz curve: n G(x) = (2/n) i=1 ((i/n) − fi ). The Gini coeﬃcient is used as a measure of wealth or income inequality: It ranges between 0 (...
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