# unbiased estimator

• May 22nd 2012, 12:45 PM
calculuskid1
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?
• May 22nd 2012, 03:33 PM
SpringFan25
Re: unbiased estimator
evaluate the expectation:

$\displaystyle E(l) = E\left((Y_1+Y_2) -Y_3 \right)$
$\displaystyle E(l) = E(Y_1+Y_2) -E(Y_3)$
$\displaystyle E(l) = E(Y_1)+E(Y_2) -E(Y_3)$
$\displaystyle E(l) = \mu_1 + \mu_2 - \mu3$
$\displaystyle E(l) = L$
• May 23rd 2012, 07:30 AM
calculuskid1
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?
• May 24th 2012, 01:28 PM
SpringFan25
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.