1. Suppose that I'm flipping a dime and quarter, such that the probability that both are head is pi, both tail is P2, only the d
ime is head is p3, and only the quarter is head is 1- P1 – P2 - p3. The entropy (or randomness) of this two-coin-flip is H(P1, P2, P3) = -pı log2 (P1) – P2 log2 (P2) – P3 log2 (P3) – (1 – P1 – P2 - P3) log2(1 – P1 – P2 - P3). Calculate a H/Op1, and evaluate it at P1 = P2 = P3 = 0.25 and at pı = 0.7, P2 = P3 = 0.1. 2. Compute the Hessian of the following functions, and verify directly that the matrix is symmetric: (a) u(x, y) = y*exy, (b) v(x, y, s) = 114,22 3. I'm climbing a mountain, whose landscape is described by the function z(x, y) = e-(x++y). I'm at position (x, y) = (-1,1). What is the maximal slope at this point? What is the slope I'm climbing if I walk southwest?
