Paper 2, Section II, 10F

Probability | Part IA, 2018

(a) Let XX and YY be independent random variables taking values ±1\pm 1, each with probability 12\frac{1}{2}, and let Z=XYZ=X Y. Show that X,YX, Y and ZZ are pairwise independent. Are they independent?

(b) Let XX and YY be discrete random variables with mean 0 , variance 1 , covariance ρ\rho. Show that Emax{X2,Y2}1+1ρ2\mathbb{E} \max \left\{X^{2}, Y^{2}\right\} \leqslant 1+\sqrt{1-\rho^{2}}.

(c) Let X1,X2,X3X_{1}, X_{2}, X_{3} be discrete random variables. Writing aij=P(Xi>Xj)a_{i j}=\mathbb{P}\left(X_{i}>X_{j}\right), show that min{a12,a23,a31}23\min \left\{a_{12}, a_{23}, a_{31}\right\} \leqslant \frac{2}{3}.

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