A shipping company handles containers in three different sizes: (1) 27 ft^3 (3 t

Question

A shipping company handles containers in three different sizes: (1) 27 ft^3 (3 times 3 times 3 times), (2) 125 ft^3, and (3) 512 ft^3, Let X_1 (i = 1, 2, 3) denote the number of type i containers shipped during a given week With mu = E (X_1) and sigma^2_1 = V (X), suppose that the mean values and standard deviations are as follows: mu_1 = 210 mu_2 = 240 mu_3 = 150 sigma_1 = 9 sigma_2 = 12 sigma_3 = 6 (a) Assuming that X_1, X_2, X_3 are independent, calculate the expected value and variance of the total volume shipped. expected value ft^3 variance ft^6 (b) Would your calculations necessarily be correct if the X's were not independent? Explain. The expected value would be correct, but the variance would not be correct. Both the expected value and the variance would be correct. Neither the expected value nor the variance would be correct. The expected value would not be correct, but the variance would be correct.

Explanation / Answer

a) E(27 X1 + 125 X2 + 512 X2)

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

= 27*210+12*240+512*150 =85350

VAR(27 X1 + 125 X2 + 512 X2)

= 27^2 VAR(X1) + 125^2 VAR(X2) + 512^2 VAR(X3)

= 27^2 * 9^2 + 125^2 * 12^2 + 512^2 *6^2

=11746233

b) if Xi s are not independent

then

option a) is correct

expectation would be correct ,but variance would not be correct

as in variance term ,cov(Xi,Xj) wil not be zero if they are dependent

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