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 out put 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 Enter prises operates at a level of production that maximizes its profit over variable cost. The possible prices facing the firm on a given day is represented by a random vari able V with R( V) = {10, 20, 30} and probability density function f( v) = .21110i(v) + .51120d v) + .3113oi(v). Input prices vary independently of output prices, and input price on a given day is the outcome of W with R(W) = {l,2, 3} and probability density function g(w) = .4J11J (w) + .3112i(w) + .3113i(w).
a. Define a random variable whose outcome repre sents Star profit over variable cost on a given day. What is the range of the random variable? What is the event space?
b. Define the appropriate probability density function for profit over variable cost. Define a probability set function appropriate for assigning probability to events relating to profit over variable cost.
c. What is the probability that the firm makes at least $100 profit over 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 over variable cost, what is the probability that it makes at least $200 profit over variable cost?