Question

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 ?i = E(Xi) and ?i2 = V(Xi), suppose that the mean values and standard deviations are as follows: ?1 = 210 ?2 = 250 ?3 = 120 ?1 = 9 ?2 = 12 ?3 = 7 (a) Assuming that X1, X2, X3 are independent, calculate the expected value and variance of the total volume shipped. [Hint: Volume = 27X1 + 125X2 + 512X3.] expected value variance answer?

Answer 1

(a)

For independent and identical variables X_i , and

The expected value of the total volume shipped = E(27X1 + 125X2 + 512X3)

E(27X1 + 125X2 + 512X3) = E(27X1) + E(125X2) + E(512X3)

= 27E(X1) + 125E(X2) + 512E(X3)

= 27(210) + 125(250) + 512(120)

= 5,670 + 31,250 + 10,240

= 47,160

The Expected value of the total volume shipped is 47,160 ft3.

For independent and identical variables X_i , and

The variance of the total volume shipped = Var(27X1 + 125X2 + 512X3)

Var(27X1 + 125X2 + 512X3) = Var(27X1) + Var(125X2) + Var(512X3)

= 27²Var(X1) + 125²Var(X2) + 512²Var(X3)

= 27²(81) + 125²(144) + 512²(49)

= 59,049 + 2,250,000 + 12,845,046

= 15,154,105 sq.ft3

The variance of the total volume shipped is 15,154,105 sq.ft3