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Math Help - unbiased estimator

  1. #1
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    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?
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  2. #2
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    Re: unbiased estimator

    evaluate the expectation:

    E(l) = E\left((Y_1+Y_2) -Y_3 \right)
    E(l) = E(Y_1+Y_2) -E(Y_3)
    E(l) = E(Y_1)+E(Y_2) -E(Y_3)
    E(l) = \mu_1 + \mu_2 - \mu3
    E(l) = L
    Thanks from calculuskid1
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  3. #3
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    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?
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  4. #4
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    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.
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