A shipping company handles containers in three different sizes: (1) 27 ft3 (3 × 3 × 3), (2) 125 ft3, and (3) 512 ft3. Let Xi (i = 1, 2, 3) denote the number of type i containers shipped during a given week. With µi = E(Xi) and σ2i = V(Xi), suppose that the mean values and standard deviations are as follows:
µ1 = 200
µ2 = 250
µ3 = 5 100
σ1 = 10
σ2 = 12
σ3 = 8
a. Assuming that X1, X2, X3 are independent, calculate the expected value and variance of the total volume shipped.
b. Would your calculations necessarily be correct if the Xi's were not independent? Explain.