By relying on arguments similar to those used in the proof of Proposition 2.10, complete the proof of Proposition 9.3.

Data From Proposition 9.3

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Proposition 9.3 For any two random variables x and x taking values in [-1, 1], the following are equivalent: (i) x1 PSD X2; (ii) fy (Fi(z) - F2(z))dz 0, for all -1 y 0x1. Moreover, if x1 FSD X2, then it also holds that x1 PSD X2. Proof The proof of the equivalence between properties (i)-(ii) is given in Exer- cise 9.1 and follows arguments similar to those used in the proof of Proposition 2.10. In view of Proposition 2.8, X FSD X2 holds if and only if Fi(x) F2(x), for every x = [-1,1]. The last claim of the proposition then follows directly from the implication (ii) (i) above.