Let

be a random sample of size n from a normal population with mean

and variance

. Assuming n=2k for some integer k, one possible estimator for

is given by:

show that

is an unbiased estimator for

Attempt:

so when I expend it out it gives:

That term in the middle is throwing me off.

I know that I eventually have to use the fact that

and

so taking that into consideration I would get:

Mr F says: What follows is wrong. See below.
I'm not sure if I should have the bottom indicator or if it should just be:

although this doesn't seem correct.

.