Star Enterprises is a small firm that produces a product that is simple to manufacture, involving only one variable input. The relationship between input and output levels is given by q = x5
, where q is the quantity of product produced and x is the quantity of variable input used. For any given output and input prices, Star Enterprises operates at a level of production that maximizes its profit over variable cost. The possible prices in dollars facing the firm on a given day is represented by a random variable V with R (V) ¼ {10,20,30} and PDF
Input prices vary independently of output prices, and input price on a given day is the outcome of W with R (W) ¼ {1,2,3} and PDF
a. Define a random variable whose outcome represents Star’s profit above variable cost on a given day. What is the range of the random variable? What is the event space?
b. Define the appropriate PDF for profit over variable cost. Define a probability set function appropriate for assigning probability to events relating to profit above variable cost.
c. What is the probability that the firm makes at least $100 profit above variable cost?
d. What is the probability that the firm makes a positive profit on a given day? Is making a positive profit a certain event? Why or why not?
e. Given that the firm makes at least $100 profit above variable cost, what is the probability that it makes at least $200 profit above variable cost?