Calculating Weighted Matrix to Reliability of Guesses

[bodensee9]

Homework Statement

Can someone help with the following? Suppose that I make guesses with errors e = -2, -1, 5 with probabilities 1/2, 1/4, 1/4. So the expected error is zero. and the variance is -2^2*1/2+-1^2*1/4+5^2/4 = 45/4.
Suppose person B makes guesses, making errors -1, 0, 1 with probabilities 1/8, 6/8, 1/8. I am supposed to find out what weights to give to my guess and the other person's guess that give reliability to both of our guesses?

Homework Equations

I think there is something in my book that the best weighted matrix should have entries where expected value of the error in bi times the value of the error in bj. So this means that the first entry in the first column should be the expected value of the error in b1 times the value of the error in b1. In this instance, would that be -2*1/2*-2? And then I would do that for the entries made up of 2 columns, -2, -1, 5 and -1, 0, -1? Thanks.

The Attempt at a Solution

 

[lippyka]

Yes, the first entry in the first column should be -2*1/2*-2. Then for the second column, it should be -2*1/4*-1, and for the third column, it should be -2*1/4*5. For the entries made up of two columns, -2, -1, 5 and -1, 0, -1, the first entry in the first row should be -2*1/8*-1, the second entry in the first row should be -2*6/8*0, and the third entry in the first row should be -2*1/8*1. The first entry in the second row should be -1*1/8*-1, the second entry in the second row should be -1*6/8*0, and the third entry in the second row should be -1*1/8*1. The first entry in the third row should be 5*1/8*-1, the second entry in the third row should be 5*6/8*0, and the third entry in the third row should be 5*1/8*1.