Examples of such distributions include the standard

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Unformatted text preview: + R2 , and the probability of a loss (including breaking even, i.e., R1 + R2 = 0) is ploss = prob(R1 + R2 ≤ 0). The goal is to find the worst-case (i.e., maximum possible) value of ploss , consistent with the following information. Both R1 and R2 have Gaussian marginal distributions, with known means µ1 and µ2 and known standard deviations σ1 and σ2 . In addition, it is known that R1 and R2 are correlated with correlation coefficient ρ, i.e., E(R1 − µ1 )(R2 − µ2 ) = ρσ1 σ2 . Your job is to find the worst-case ploss over any joint distribution of R1 and R2 consistent with the given marginals and correlation coefficient. We will consider the specific case with data µ1 = 8, µ2 = 20, σ1 = 6, 56 σ2 = 17.5, ρ = −0.25. We can compare the results to the case when R1 and R2 are jointly Gaussian. In this case we have 2 2 R1 + R2 ∼ N (µ1 + µ2 , σ1 + σ2 + 2ρσ1 σ2 ), which for the data given above gives ploss = 0.050. Your job is to see how much larger ploss can possibly be. This is an infinite-dimensional optimization pr...
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