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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 worstcase (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 worstcase 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 inﬁnitedimensional optimization pr...
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 Fall '13
 F.Borrelli
 The Aeneid

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