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Thread: Expected Value

  1. #1
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    Expected Value

    For a sample of identically and independently distributed variables , each having a mean μ and variance $\displaystyle \sigma^2$

    for $\displaystyle \subscript{i}=1, 2, ...., n$

    what is
    i) $\displaystyle
    E(X_i \bar{X}), where \bar{X}= (\sum{X_i})/n
    $

    ii) $\displaystyle
    E(\bar{X}^2)
    $

    iii) $\displaystyle E[\bar{X}(X_i-\bar{X})]$

    thanks a lot
    Last edited by cky78; Sep 23rd 2010 at 06:54 AM.
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  2. #2
    Grand Panjandrum
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    Quote Originally Posted by cky78 View Post
    For a sample of identically and independently distributed variables , each having a mean μ and variance $\displaystyle \sigma^2$

    for $\displaystyle \subscript{i}=1, 2, ...., n$

    what is
    i) $\displaystyle
    E(X_i \bar{X}), where \bar{X}= (\sum{X_i})/n
    $

    ii) $\displaystyle
    E(\bar{X}^2)
    $

    iii) $\displaystyle E[\bar{X}(X_i-\bar{X})]$

    thanks a lot
    Because $\displaystyle X_i, \ X_j , \ i\ne j$ are independent $\displaystyle E(X_iX_j)=\mu^2$ (just write out the expectation of the product and observe that independence means that $\displaystyle p(x_i,x_j)=p(x_i)p(x_j) $)

    Then the rest should follow easily

    (you might also need $\displaystyle E(X_iX_i)=\sigma^2+\mu^2$ )

    CB
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