Problem 1 (25 points): Investing A model for the movement of a nancial index (a quantity whose value changes over time) assumes that, given the present value V\" of the index at time slot 11, n = 0, 1, 2, . . ., after one time slot it will be Vn+1 = U - V\" with probability p or Vn+1 = D - Vn with probability 1 p. Assume that successive movements of the value are independent, U = 1.01, D = 0.99, p = 0.51, and V6 is known. (Note: Even though I am giving you specic values for quantitates, you do not need to make numerical computations once you express things as a function of those numerical values.) (i) (4 points) Let Xn be a random variable dictating whether the index will go up or down after time slot 12, that is, Vn+1 = Xn - V\". Express Xn as a function of U and V and compute its mean and variance. (ii) (5 points) Write the value of the index at time n, V\