Suppose that for r.v.’s X₁ and X₂ with respective d.f.’s F₁(x) and F₂(x), it is true that F₁(x) ≤ F₂(x) for all x’s. Such a relation is referred to as the first stochastic dominance (FSD), and we say that X₁ (or its d.f. F₁) dominates X₂ (respectively, F₂) in the sense of the FSD.

(a) Argue that if we follow the rule “the larger, the better” (for example, if the X’s are r.v.’s of a future income), then the r.v. X₁ is preferable. 

(b) In particular, argue that one should expect that E{X₁} ≥ E{X₂}. Prove it. (Advice: You may appeal to (6.2).) 

(c) Show that the FSD relation is true, for example, if X₁ and X₂ are uniform on [0,a₁] and [0,a₂], respectively, and a₁ ≥ a₂; or if X₁ and X₂ are exponential with respective parameters a₁ and a₂, and a₁ ≤ a₂. In both cases, provide the corresponding graphs.