One way retail industry uses machine learning is to predict how much quantity Q of some product they should buy to maximize their profit. The optimal quantity depends on how much demand D there is for the product, as well as its cost for the retailer to buy C, and its selling price P to the customer. Assuming that the demand D is distributed as P(D), we can evaluate the expected profit considering two cases: if D Q, then the retailer sells all Q items and makes a profit = (P C)Q. but if D < Q, then the retailer can only sell D items at profit (P C)D, but has lost C(Q D) on unsold items. (i) What is the expected profit if the retailer buys Q items? Simplify the expression as much as possible. (ii) By taking the derivative (with respect to Q) of the above expression for expected profit, show that the optimal quantity Q to buy satisfies Q = F 1 (1 (C/P)), where F is the CDF of D. That is, the optimal Q is when the cumulative density (of D) equals 1 (C/P)