On a busy interstate highway, I there are 21 cars in a particular lane. Let X1... X20 denote the 20 distances between these 21 cars (i.e., X1 is the distance between the first and second cars; X2 is the distance between the second and third cars; etc.). At a particular moment, the Xj's are judged to be approximately Normal, with an average of 500 feet between consecutive cars and standard deviation of 75 feet. Find the probability that the row of 21 cars is less than two miles long (each mile contains 5280 feet) if:
a. The length of each car is assumed to be negligible, i.e., we do not take I into account the lengths of the 21 cars themselves.
b. The length of each car is assumed to be fixed, i.e., 13.5 feet long. Thus, we want the probability of
X1 + . . . + X20 + (13.5) (21) < (5280) (2)
c. The lengths of the cars are assumed to be independent and normally distributed too, each with average length 13.5 feet and standard deviation 1 foot. In this case, if Y1... Y21 are the length of the cars Thus, we want the probability of
X1 + . . . + X20 + Y1 + . . . + Y21 < (5280) (2)