The daily quantity demanded of unleaded gasoline in a regional market can be represented as \(Q=100-10 p+E\), where \(p \in[0,8]\), and \(E\) is a random variable having a probability density given by \(f(e)=0.025 I_{[-20,20]}(e)\).

Quantity demanded, \(Q\), is measured in thousands of gallons, and price, \(p\), is measured in dollars.

a. What is the probability of the quantity demanded being greater than 70,000 gal if price is equal to \(\$ 4\) ? if price is equal to \(\$ 3\) ?

b. If the average variable cost of supplying \(Q\) amount of unleaded gasoline is given by \(C(Q)=Q^{.5} / 2\), define a random variable that can be used to represent the daily profit above variable cost from the sale of unleaded gasoline.

c. If price is set equal to \(\$ 4\), what is the probability that there will be a positive profit above variable cost on a given day? What if price is set to \(\$ 3\) ? to \(\$ 5\) ?

d. If price is set to \(\$ 6\), what is the probability that quantity demanded will equal 40,000 gal?