The weekly average price, in dollars per gallon, and quantity sold of organic milk, measured in thousands of gallons, in a large west coast market in the fall is represented by the outcomes of a bivariate random variable \((P, Q)\) having a multivariate normal distribution, \(N(\boldsymbol{\mu}, \mathbf{\Sigma})\) with the following mean vector and covariance matrix:

\(\boldsymbol{\mu}=\left[\begin{array}{c}3.50 \\ 100\end{array}ight]\) and \(\boldsymbol{\Sigma}=\left[\begin{array}{cc}.01 & -.7 \\ -.7 & 100\end{array}ight]\)

(a) Define an interval event, centered at the mean, that will contain the outcomes of the price of organic milk with .95 probability.

(b) What is the probability that quantity sold will exceed 110,000 gallons?

(c) Define the moment generating function for \((P, Q)\). Use it to define the expected value of weekly total sales of organic milk, in dollars.

(d) Define the regression function of quantity sold as a function of the price of organic milk. Does the regression function make economic sense? Use the regression function to find the expected quantity sold given a price of \(\$ 3.40\).

(e) Can the bivariate distribution of price and quantity actually be multivariate normally distributed in this case, or must the normal distribution be only an approximation? Explain.