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Math Help - Covariance question

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    Covariance question

    please please help me out for these question below. They are very important to me. As I am not good on Statistic and I don't have the model answers for them, I really want someone who can show me the answers about them. Therefore, I can check if i was doing the questions in the right way.

    Thank you very much!!!

    ================================================== =======
    Question 1 -- Let X, Y and Z be random variables.
    (a) Define the covariance Cov(X, Y ).
    (b) Prove that Cov(Z,X + Y ) = Cov(Z,X) + Cov(Z, Y ).
    (c) Suppose that X and Y are independent, with Var(X) = 25 and Var(Y ) = 6,
    and that Z = X + 2Y . Find the correlation between X and Z.
    Last edited by mr fantastic; April 30th 2010 at 04:09 PM. Reason: Re-titled.
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  2. #2
    MHF Contributor harish21's Avatar
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    Quote Originally Posted by gklove56 View Post
    please please help me out for these question below. They are very important to me. As I am not good on Statistic and I don't have the model answers for them, I really want someone who can show me the answers about them. Therefore, I can check if i was doing the questions in the right way.

    Thank you very much!!!

    ================================================== =======
    Question 1 -- Let X, Y and Z be random variables.
    (a) Define the covariance Cov(X, Y ).
    (b) Prove that Cov(Z,X + Y ) = Cov(Z,X) + Cov(Z, Y ).
    (c) Suppose that X and Y are independent, with Var(X) = 25 and Var(Y ) = 6,
    and that Z = X + 2Y . Find the correlation between X and Z.

    (a) Cov(X,Y) = E[({X_1}-{\mu_x})({Y}-{\mu_y})]

    = E[XY] - {\mu_y}E(X) - {\mu_x}E(Y) + {\mu_x}{\mu_y}

    =E[XY] - {\mu_y}{\mu_x}- {\mu_x}{\mu_y}+{\mu_y}{\mu_y}

    \therefore Cov(X,Y)= E[XY] - {\mu_x}{\mu_y} = E[XY] - E[X]E[Y]


    After knowing (a), you can solve (b) as:

    Cov(Z,X+Y) = E[Z(X+Y)] - E[Z]E[X+Y]

    = E[ZX]+E[ZY] - E[Z] {E[X]+E[Y]}

    = E[ZX]+E[ZY] - E[Z]E[X] - E[Z]E[Y]

    = \{E[ZX]-E[Z]E[X]\} + \{E[ZY] - E[Z]E[Y]\}

    = Cov(Z,X) + Cov(Z,Y)

    (c) refer to the definition of correlation( \rho) in your book/notes
    Last edited by harish21; April 30th 2010 at 11:16 PM.
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    Thank you so much, you are great!!!
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