A shipping company handles containers in three different sizes: (1) 27 ft3 (3 ×

Question

A shipping company handles containers in three different sizes: (1) 27 ft3 (3 × 3 × 3), (2) 125 ft3, and (3) 512t3. Let X, (i = 1, 2, 3) denote the number of type i containers shipped during a given week. With .EM and o? =v(x), suppose that the mean values and standard deviations are as follows: 1.210 2.240 3.130 1 = 10 2 = 13 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] ft3 ft6 expected value variance (b) Would your calculations necessarily be correct if the X/s were not independent? Explain O The expected value would be correct, but the variance would not be correct. O The expected value would not be correct, but the variance would be correct. O Both the expected value and the variance would be correct. O Neither the expected value nor the variance would be correct.

Explanation / Answer

Part A ) Here x1 , x2 and x3 are independent.

E( xc) = c E(X) where c = constant and V(xc) = c2 V(x)

V( X+Y) = V(X )+ V( Y) if X and Y are independent random variables

E( X+Y) = E(X) + E(Y)   if X and Y are independent random variablesion

Given equation

27 X1 + 125 X2 + 512 X3 by using above propertise

Then E(27 X1 + 125 X2 + 512 X3 ) = 27 E(X1) + 125 E(X2) + 512 E(X3) = 27 *210 + 125*240 + 512*130 = 102230

V( 27 X1 + 125 X2 + 512 X3 ) = 272 V(x1) + 1252 V(X2) + 5122 V(X3) = 153090 +3750000+34078720 =37981810

Part B) If Xi are independent then  

The expected value would be correct , butb the variance would not be correct.

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