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Math Help - Expected Value of nS^2

  1. #1
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    Expected Value of nS^2

    I am curious if any expert knows what the expected value is of SUM (Xi - X-bar)^2 from 1 to n. Where Xi are a random sample from a distribution?

    We have been showing that the expected value of S^2 is sigma squared. But I am wondering how to do this without the n or (n-1) divisor.

    We were able to do the formers by re-writing S^2 as SUM(Xi^2)^2 /n - x-bar^2

    but this does not work for SUM (Xi - X-bar)^2

    Thanks!
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  2. #2
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    Quote Originally Posted by B_Miner View Post
    I am curious if any expert knows what the expected value is of SUM (Xi - X-bar)^2 from 1 to n. Where Xi are a random sample from a distribution?

    We have been showing that the expected value of S^2 is sigma squared. But I am wondering how to do this without the n or (n-1) divisor.

    We were able to do the formers by re-writing S^2 as SUM(Xi^2)^2 /n - x-bar^2

    but this does not work for SUM (Xi - X-bar)^2

    Thanks!
    You know that S^2 = \frac{\sum_{i=1}^n (x_i - \overline{x})^2}{n-1} is an unbaised estimator of \sigma^2. Therefore:


    E\left( \frac{\sum_{i=1}^n (x_i - \overline{x})^2}{n-1} \right) = \sigma^2


    \Rightarrow E\left(\sum_{i=1}^n (x_i - \overline{x})^2 \right) = (n - 1) \sigma^2.
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