This exercise concerns the VaR criterion with a parameter . R.v.s X belowwith or without indicescorrespond to an income. (a) Let a r.v. X 1

This exercise concerns the VaR criterion with a parameter γ. R.v.’s X below—with or without indices—correspond to an income.

(a) Let a r.v. X₁ take on values 0,1,2,3 with probabilities 0.1, 0.3, 0.4, 0.2, respectively, and a r.v. X₂ take on the same values with probabilities 0.1, 0.4, 0.2, 0.3. Find all γ’s for which X₁ ≳ X₂.

(b) Let X₁ be uniform on [0,1], and X₂ be exponential with E{X₂} = m. When is the relation X₂ ≳ X₁ true for all γ’s? Let m = 0.98. Find all γ’s for which X₂ ≳ X₁.

(c) Let r.v.’s X₁ = 1− ξ₁, X₂ = 1− ξ₂, where the loss ξ₁ is uniform on [0,1], and ξ₂ is exponential with E{ξ₂} = m. When is the relation X₁ ≳ X₂ true for all γ’s? Let m = 0.98. Find all γ’s for which X₁ ≳ X₂.

(d) In Exercise 1c, let ξ₁ be uniform on [0,3], and let ξ₂ be uniform on [1,2]. Find all γ’s for which X₁ ≳ X₂.