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Math Help - Dependent Variance ?

  1. #1
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    Dependent Variance ?

    I was hoping someone can show me the proof for
    V(y1-Y2) = V(y1)+V(y2)- 2 Cov(Y1,Y2)
    or
    V(y1+Y2) = V(y1)+V(y2)+ 2 Cov(Y1,Y2)

    I understand the proof for V(y1+Y2) = V(y1) + V(Y2) for independent but im having trouble inferring for the dependent.
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  2. #2
    Grand Panjandrum
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    Quote Originally Posted by Chowder610 View Post
    I was hoping someone can show me the proof for
    V(y1-Y2) = V(y1)+V(y2)- 2 Cov(Y1,Y2)
    or
    V(y1+Y2) = V(y1)+V(y2)+ 2 Cov(Y1,Y2)

    I understand the proof for V(y1+Y2) = V(y1) + V(Y2) for independent but im having trouble inferring for the dependent.
    V(y_1+y_2)=E( [(y_1+y_2)-\overline{(y_1-y_2)}]^2)

    but

     <br />
\overline{(y_1-y_2)}=\overline{y_1} + \overline{y_2}<br />

    so:

    V(y_1+y_2)=E( [ (y_1-\overline{y_1}) +(y_2-\overline{y_2})]^2)  =E((y_1-\overline{y_1}) ^2) + 2E((y_1-\overline{y_1}) (y_2-\overline{y_2}) )+E((y_2-\overline{y_2}) ^2)

    CB
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  3. #3
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    Thank you
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