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.]

Step-by-step explanation:

Hello!

To know what will be the resulting E(X) and V(X) of the total volume shipped, you have to add the three variables.

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!

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