Question

The scale in Beatrice's bathroom gives readings which are normally distributed, with mean equal to the weight of the object placed on it, and variance σ², which is unknown, and does not depend on the weight. Beatrice wishes to estimate σ². She places on the scale an object whose weight is known to be exactly 50 lbs, getting a reading Y₁. She repeats this independently n - 1 times, getting readings Y₁, Y₂,...,Y_n. She is considering the following two estimators of σ²: 

Her reason for considering the second estimator is that she thinks that since she knows the true weight, she might as well use it, to get more accuracy. Which estimator is better, and does the answer depend on what is the true value of σ?

Answer 1

Estimator A is better as it is unbiased while the estimator B is biased. here is a proof of why A is an unbiased estimatormof population variance-

Thanks!