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Math Help - the expectation of the sum of yi minus ybar squared

  1. #1
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    the expectation of the sum of yi minus ybar squared

    In general what is the expectation of the sum of yi minus ybar squared

    E(\sum( y_i - \bar{y})^2) =  E(\sum( y_i^2 - 2\bar{y}y_i + \bar{y}^2)) =

    E(\sum y_i^2 -\sum \bar{y}^2)) = \sum E(y_i^2) -\sum E(\bar{y}^2)

    is this the same as the sum of of VAR(Y)?
    Last edited by kingsolomonsgrave; November 5th 2013 at 12:48 AM.
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  2. #2
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    Re: the expectation of the sum of yi minus ybar squared

    Hey kingsolomonsgrave.

    What do you mean by sum of Var[Y]? Can you write it out in terms of sigma notation and expectation/variance of random variables?
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  3. #3
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    Re: the expectation of the sum of yi minus ybar squared

    var(Y) in terms of expectation is E (Y_i - \bar{Y)}^2 = E(Y^2)-[E(y)]^2

    and

     E(\sum (Y_i -\bar{Y})^2) = E[\sum(Y_i^2 -2Y_i\bar{Y} +\bar{Y})]

     =\sum[E(Y_i ^2) - [E(\bar(Y))^2] = \sum(VAR(Y)) = nVAR(Y)
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  4. #4
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    Re: the expectation of the sum of yi minus ybar squared

    This looks right and remember also that expectation is linear so if you have a sigma inside the expectation, you can take it outside the expectation as well.
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