A shipping company handles containers in three different sizes: (1) 27 ft (3 x 3 x 3), (2) 125 ft', and (3) 512 ft. Let X, (i = 1, 2, 3) denote the number of type i containers shipped during a given week. With , = E(X) and =V(X), suppose that the mean values and standard deviations are as follows: = 200 = 10 M = 250 = 12 3 = 100 = 8

a. Assuming that X1, X2, X3 are independent, calculate the expected value and variance of the total volume shipped. [Hint: Volume = 27X + 125X+ 512X3.]

b. Would your calculations necessarily be correct if the X's were not independent? Explain.