A surveyor measures an angle with two instruments; call them A and B. Both instruments produced unbiased measurements, but the standard deviation of measurements made with instrument A is twice that of B. Due to environmental effects (wind, thermal expansion, etc) simultaneous measurements made with the two instruments have a correlation of r=0.5.
Let X and Y be random variables representing simultaneous measurements of a bearing made with instruments A and B, respectively. You are going to use a linear combination of the two measurements as your estimator (call it S) of the true bearing (call it s):
S=aX+bY
Where a and b are predetermined weights.
(a)What conditions must be placed on the weights a and b so that E[S]=s
That is, so that S is an unbiased estimator of s?
(b)What values of a and b will minimize the variance of S? The answer to this question will take the from a equals a specific number and b equals a specific number.