A shipping company handles containers in three different sizes:
(a) 27 ft³ (3×3×3), (b) 125 ft³, and (c) 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 σi2 = V(Xi), suppose that the mean values and standard deviations are as follows:

μ1 = 200; μ2 = 240; μ3 = 120
σ1 = 10; σ2 = 13; σ3 = 8.

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

According to the properties of the E(X) The expected value of two variables or more independent variables is equal to the addition of the expected values of each variable: X+Y⇒ E(X+Y)= E(X) + E(Y)

In this case:

E(X₁+X₂+X₃)= E(X₁)+E(X₂)+E(X₃)= 200 + 240 + 120= 560 ft³

The same happens with the variance, if you add two independent variables, the variance of the resulting variable will be equal to the addition of the variance of each variable: X+Y ⇒ V(X+Y)= V(X)+V(Y)

In this case:

V(X₁+X₂+X₃)= V(X₁)+V(X₂)+V(X₃)= 10+13+8= 31

I hope it helps!