7. The quantity of wheat demanded, per day, in a midwestern market during a certain marketing period is represented by Q ¼ 100; 000 12; 500 P þ V for p 2 ½  2; 6 ;

where Q is quantity demanded in bushels;

p is price/bushel; and V is approximately normally distributed.

You know that the expected quantity demanded is given by EðQÞ ¼ 100; 000 12; 500 p for p 2 ½  2; 6 ;

and thus is a function of p, and the variance of quantity demanded is var(Q) ¼ 16 106

.

(a) What is the mean and variance of V?

(b) If p ¼ 4, what is the probability that more than 50,000 bushels of wheat will be demanded?

(c) If p ¼ 4.50, what is the probability that more than 50,000 bushels of wheat will be demanded?

(d) For quantity demanded to be greater than 50,000 bushels with probability .95, what does p have to be?

(e) Is it possible that V could actually be normally distributed instead of only approximately normally distributed? Explain.