I'm confused as to the covariance of Y's being
that means the correlation is 1.
The variance is just
I want to understand the calculation of the variance of a sum of rvs where each variable is weighted using a constant coefficient. I post the full question and answer here and I highlight the areas that I am struggling to follow.
Question.
is a sample from a population with mean and variance .
The sample is not random, for . Let .
(a) Give the condition on a constants for U to be unbiased estimator of .
(b) Under this condition, calculate MSE(U).
Answer.
(a) is not a problem, just need to calculate a if which gives
and is the condition.
(b) given (a) is met, then MSE(U)=Var(U):
(1)
and starting from (2) I find it tricky to follow the solution:
(2)
(3)
It looks to me like a sum of all entries of the variance matrix of an nx1 vector... but I cannot yet move beyond that.
Then it gets OK again, a straightforward substitution
(4).
I'd appreciate a bit of clarification on manipulations in (2) and (3)...
I've just realised I messed up the final answer too, it should read:
so apologies, my bad.
Anyways, I had a bright idea to look up "variance of a sum of correlated variables" and I've figure it out
(per Wiki),
which also includes when i=j:
(we can take constants outside)
(substibute info given in the question)
Finally, taking common factors outside the summation sign
as per the final book answer.
Strange though that I didn't use the fact that from (a) that