All r v s appearing below are defined on the probability space ((, A, P) (i) If X ( 0, then X ( 0 and X 0 only if P (X 0) 1 (ii) Let X ( Y with finite expectations Then X ( Y and X Y only if P(X Y) 1 (iii) Let X Y with finite expectations Then X Y (iv) Let g I open subset of ( be strictly convex (i e ,) g ax (1 a)x ' ag(x) (1 a) g (x ') x, x ' ( I, 0 a 1), let Z be a r v taking values in I, let Z ( I, and let g(Z) exits Then g(Z) g(Z) unless P(Z constant) 1

Question: All r.v.s appearing below are defined on the probability space ((, A, P). (i) If X ( 0, then X ( 0 and

All r.v.s appearing below are defined on the probability space ((, A, P).
(i) If X ( 0, then ε X ( 0 and ε X = 0 only if P (X = 0) = 1.
(ii) Let X ( Y with finite expectations. Then ε X ( εY and εX = εY only if P(X = Y) = 1.
(iii) Let X > Y with finite expectations. Then εX > εY.
(iv) Let g : I open subset of ( be strictly convex (i.e.,)
g[ax + (1 - a)x'] < ag(x) + (1 - a) g (x').
x, x' ( I, 0 < a < 1), let Z be a r.v. taking values in I, let εZ ( I, and let εg(Z) exits. Then εg(Z) > g(εZ) unless P(Z = constant) = 1.

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