The daily price, \(p\), and quantity, demanded, \(q\), of gasoline on a European Wholesale spot market can be viewed (approximately) as the outcome of a bivariate normal random variable, where the bivariate normal density has mean vector and covariance matrix as follows:

\(\mu=\left[\begin{array}{c}2.50 \\ 100\end{array}ight], \Sigma=\left[\begin{array}{cc}.09 & -1 \\ -1 & 100\end{array}ight]\)

The price is measured in U.S. dollars per gallon of gasoline, and the quantity demanded is measured in thousands of gallons.

(a) What is the probability that greater than 110,000 gallons of gasoline will be demanded on any given day?

(b) What is the probability that the price of gasoline will be between \(\$ 2.00\) and \(\$ 3.00\) on any given day?

(c) Define the regression function of \(Q\) on \(p\). Graph the regression function. What is the expected daily quantity of gasoline demanded given that price is equal to \(\$ 3.00\) ?

(d) Given that the price equals \(\$ 3.00\), what is the probability that quantity demanded will exceed 110,000 gallons? What is this probability on a day when price equals \(\$ 2.00\) ?