For the case of the TailVaR criterion, solve

(a) Exercise 1a;

(b) Exercise 1d.

Exercise 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.

(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.