A premium principle is said to be sub-additive if for two risks X1 and X2 (which may be dependent), x,+x, x, + IIx,. Under what conditions is the variance principle sub-additive? The mean value principle states that the premium, x, for a risk X is given by x v(E[v(X)]) = where v is a function such that v'(x) > 0 and v"(x) 0 for x > 0. (a) Calculate x when u(x) = x and X ~ y(2, 2). (b) Construct a counter example to show that this principle is not consistent.