Question: A triangle has angles A, B, and C (in degrees). The angles are estimated, using a certain device, giving independent estimators XA, XB, and XC,

A triangle has angles A, B, and C (in degrees). The angles are estimated, using a certain device, giving independent estimators XA, XB, and XC, which are all normally distributed,with the correct mean, and standard deviation equal to 1, i.e. XA ~ N(A, 1), XB ~ N(B, 1), XC ~ N(C, 1).

For a certain reason, interest lies exclusively in angle A.

Maurice wants to estimate A by using XA.

Alphonse wants to be clever and says that A should be estimated by 180 - XB - XC.

Nadine wants to use the average Y =1/2XA +1/2(180 - XB - XC).

A) What are the expected values of each of the three estimators?

B) What are the variances of each of the three estimators? From this, state which is the best estimator and which is the worst? Give a complete ranking.

C) What are the distributions of each of the three estimators?

D) Consider the generalization of Nadines estimator given by the weighted average Yw = w*XA + (1 - w)(180 - XB - XC), where w ? [0, 1]. Which value of w is best?

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