Exercise 5.4 Consider two random variables Xi, i = 1, 2, with distribution functions Fi(x). If F1(x) ≤ F2(x) for all x ∈ IR, then X1 is said to be greater than X2 in the sense of first-order stochastic dominance, denoted by X1 ≥st X2. Prove the following

(1) If X1 ≥lr X2 then X1 ≥st X2.
(2) If X1 ≥st X2 then E[f(X1)] ≥ E[f(X2)] for any non-decreasing function f(x), for which the expectations exist. Hence, in particular, X1 ≥st X2 implies E[X1] ≥ E[X2].
(3) Conversely, if E[f(X1)] ≥ E[f(X2)] for any non-decreasing function f(x), for which the expectations exist, then X1 ≥st X2.