1. ## unbiased estimator

So I have 3 independent populations with μ1, μ2, and μ3.
It is given to calculate their average we use L= (
μ1+μ2) - μ3

We have random samples for each population with respective sample mean Y1, Y2, and Y3.

Then we get the estimator l= (Y1 + Y2) - Y3
. _
So the Question is: Is l an unbiased estimator for L?

I know in order to be an UE the expected value of the estimator has to be equal to the actual value, but how do we show this?

2. ## Re: unbiased estimator

evaluate the expectation:

$\displaystyle E(l) = E\left((Y_1+Y_2) -Y_3 \right)$
$\displaystyle E(l) = E(Y_1+Y_2) -E(Y_3)$
$\displaystyle E(l) = E(Y_1)+E(Y_2) -E(Y_3)$
$\displaystyle E(l) = \mu_1 + \mu_2 - \mu3$
$\displaystyle E(l) = L$

3. ## Re: unbiased estimator

wow that is very simple!
One more question, if each population has the same population standard deviation σ=5 and each sample is of size n=10.
Compute the variance of the estimator l. That is, compute σ^2{l}.

From class I learnt that σ^2{l}=σ^2/n, but how do I use that to solve it?

4. ## Re: unbiased estimator

Y1,y2,y3 are independent so the variance of l is just the sum of the variances of Y1...Y3.

you can find teh variance of Y1...Y3 using the formula you posted.