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Math Help - 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  \sigma^2

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

    what is
    i)  <br />
E(X_i \bar{X}), where \bar{X}= (\sum{X_i})/n<br />

    ii)  <br />
E(\bar{X}^2)<br />

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

    thanks a lot
    Last edited by cky78; September 23rd 2010 at 07:54 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  \sigma^2

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

    what is
    i)  <br />
E(X_i \bar{X}), where \bar{X}= (\sum{X_i})/n<br />

    ii)  <br />
E(\bar{X}^2)<br />

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

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

    Then the rest should follow easily

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

    CB
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