Chapter 5.5, Problem 58E

A shipping company handles containers in three different sizes: (1) 27 ft³ (3 × 3 × 3), (2) 125 ft³, and (3) 512 ft³. Let X_i (i = 1, 2, 3) denote the number of type i containers shipped during a given week. With µ_i = E(X_i) and σ_i² = V(X_i), suppose that the mean values and standard deviations are as follows:

      $\begin{array}{lllllllll}{\mu }_1\hfill & =200\hfill & {\mu }_2\hfill & =250\hfill & {\mu }_3\hfill & =100\hfill & \hfill & \hfill & \hfill \\ {\sigma }_1\hfill & =10\hfill & {\sigma }_2\hfill & =12\hfill & {\sigma }_3\hfill & =8\hfill & \hfill & \hfill & \hfill \end{array}$

1. a. Assuming that X₁, X₂, X₃ are independent, calculate the expected value and variance of the total volume shipped. [Hint: Volume = 27X₁ + 125X₂ + 512X₃.]
2. b. Would your calculations necessarily be correct if the X_i’s were not independent? Explain.