Here is a puzzle in probability. Two navigators estimate the distance?

Question

of their ship from the shore and come up with values E1 and E2. In the past they are known to have estimated distances with standard deviations of S1 and S2. What should the captain of the ship do to combine these two values to estimate the distance from shore.

Any detailed explanation and/or links would be helpful.

Thanks.

Answer 1

The roots of this problem lie in the fundamental problem addressed by kalman filters. You want to read up about kalman filters to fully understand the solution.

Here is the approach...

given two estimates of a value that is normally distributed, you assume that a better estimate exists with value E = wE1 + (1-w)E2. The variance of this new estimate is given by S^2 = w^2.S1^2 + (1-w)^2.S2^2. You need to differentiate the second equation, look for a minima and plug that value of the weight into the first equation to get your optimal estimate. Hope this helps.

Answer 2

If you are looking for an over-all average estimate, add all the estimates up and divide by the number of estimates to arrive at an average sum.
assign values to see

E1=150mi; E2=153mi; S1=149.5mi; S2=156.3mi

150+153+149.5+156.3=608.8/4=152.2mi, or about 152mi.

Answer 3

Since the one with lower standard deviation is more accurate in the long run, his answer should be weighted more than the other when averaging. You could add the two SD's, multiply the more accurate guy's by the larger SD and the less accurate guy's by the smaller SD, then divide by the sum of the SD's. How about that?