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 \(q=x^{.5}\), 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 \(R\) \((V)=\{10,20,30\}\) and PDF

\(f(v)=.2 I_{\{10\}}(v)+.5 I_{\{20\}}(v)+.3 I_{\{30\}}(v)\).

Input prices vary independently of output prices, and input price on a given day is the outcome of \(W\) with \(R\) \((W)=\{1,2,3\}\) and PDF

\(g(w)=.4 I_{\{1\}}(W)+.3 I_{\{2\}}(W)+.3 I_{\{3\}}(W)\).

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?