Question

A retail store is currently short of a certain product, and the manager is now planning on how many amounts of such product to import for the sales of them in next month. The store pays $20 (the import price) per unit of the product, and then sells it for $30 (the marked price in that store) per unit. However, if the store is not able to sell these imported products in the next month, these products must be disposed and can not be sold afterwards. Therefore, there is a trade-off between maximizing the profits (the total amount of $ gained by selling the products minus the total amount of $ paid to import the products) and minimizing the risk of overstock (which will then result in wasting $20 per unit of the unsold products). Suppose that the demand for such products in a randomly chosen month has the Binomial distribution Binom(n = 20, p = 0.3). Note that it is possible that the demand exceeds the amounts of the products in the store, in which case the store can only sell all the amount of the products in the store.

(a) Let A (a non-negative integer) denote the amount of such products that the manager imports, and let B (a non-negative integer) denote the demand of such products. Write down the expression of the profit (in $) that the store can make in next month, which should involve A and B.

(b) Let A (a non-negative integer that is not greater than 20) denote the amount of such products that the manager imports. Find the expression of the conditional probability that the demand of such products is k, given that the demand does not exceed A. You may assume here that k is some non-negative integer that is not greater than A. Your answer should involve A and k.

(c) Suppose the manager finally decides to import A units of such products (you may assume that A ≤ 20), find the expected profit that the retail store can make for selling these products in next month. Your answer should involve A.

Answer 1