Problem 1: A store has a featured new seasonal product

Problem 1:

A store has a featured new seasonal product. Let N be the random variable which designates the number of customers who come to the store during the season, where N∼Poi (A). We estimate the probability that a customer will buy this new product is B and is independent from customers to customer.

1. We suppose that the store has unlimited stock of this product. Let the random variables X and Y such that X = the number of customers who buy the product; Y = the number of customers who do not buy the product. Are the X and Y variables independent? Justify.
2. The store has a profit of C$ for each unit sold. Each unsold unit should be in storage for the next year at a cost of D$.
   Determine the value of the number of stored units “n” the store should have to maximize its average profit.

Note that A = 22, B = 0.566, C = 13, D = 7.

If necessary and depending on the context, use a critical threshold of 5%, or a confidence level of 95%.