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Math Help - Expected Value and Variance (Basic Properties)

  1. #1
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    Expected Value and Variance (Basic Properties)

    Hey guys, Im going over my Econometrics homework () and was hoping for some help to confirm or correct my answers. Its regarding the basic properties of Variance and Expected Value. Any help is much appreciated. Here's what I got so far (my answers in italicized font):

    1) Calculate E(X + Y) = muX + muY
    2) Calculate V(X + Y) = sigmaX + sigmaY + 2sigmaXY
    3) Calculate E(2X) = 2muX
    4) Calculate E(2Y) = ???
    5) Calculate V(2X + 3Y) = 4sigmaX + 9sigmaY + 12sigmaXY
    6) Calculate E(1 + 2X) = 2muX + 1
    7) Calculate V(1 + 2X) = 4sigmaX
    8) Calculate E(3X + 4(Y + 1)) = 3muX + 4muY + 4
    9) Calculate V(3X + 4(Y + 1)) = 9sigmaX + 16sigmaY
    10) Calculate E(aX + b(Y + c)) = amuX + bmuY + bc
    11) Calculate V(aX + b(Y + c)) = a^2sigmaX + b^2sigmaY
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  2. #2
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    Hello,
    Quote Originally Posted by RutgersGirl View Post
    Hey guys, Im going over my Econometrics homework () and was hoping for some help to confirm or correct my answers. Its regarding the basic properties of Variance and Expected Value. Any help is much appreciated. Here's what I got so far (my answers in italicized font):
    Does sigma stand for the variance ? Because all the sigma's you've used should be variances
    And usually, sigma stands for standard deviation.

    1) Calculate E(X + Y) = muX + muY

    2) Calculate V(X + Y) = sigmaX + sigmaY + 2sigmaXY
    No, the formula is V(X+Y)=V(X)+V(Y)+2Cov(X,Y)
    where Cov denotes the covariance. Note that Cov(X,Y)=0 if X and Y are independent.
    Cov(X,Y)=E[(X-mu(X))(Y-mu(Y))=E(XY)-E(X)E(Y) (the red one is the most commonly used)
    3) Calculate E(2X) = 2muX

    4) Calculate E(2Y) = ???
    Exactly the same as above in 3)
    For any random variable X (or Y, or whater your variable is called), E(aX)=aE(X)
    5) Calculate V(2X + 3Y) = 4sigmaX + 9sigmaY + 12sigmaXY
    There is still a problem with the 12sigma(XY), otherwise, it's correct.
    6) Calculate E(1 + 2X) = 2muX + 1

    7) Calculate V(1 + 2X) = 4sigmaX

    8) Calculate E(3X + 4(Y + 1)) = 3muX + 4muY + 4

    9) Calculate V(3X + 4(Y + 1)) = 9sigmaX + 16sigmaY
    Once again, look at the covariance. Don't hesitate to ask if you don't know what to do.
    10) Calculate E(aX + b(Y + c)) = amuX + bmuY + bc

    11) Calculate V(aX + b(Y + c)) = a^2sigmaX + b^2sigmaY
    Once again, the formula for the variance is false



    Good working anyway, it's good to see someone's attempts :P
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    Wow! Thanks for all your help. I was able to get the homework down, but I didn't do as good as I was hoping on the exam. Much appreciated!
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  4. #4
    MHF Contributor matheagle's Avatar
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    Her sigma_xy IS covariance in most books.
    So that's just notation.
    The only thing I see wrong is number 11.
    You're missing the covariance term.
    Instead of
    Calculate V(aX + b(Y + c)) = a^2sigmaX + b^2sigmaY
    you need to add 2ab times the covariance of X and Y.
    The constant term bc disappears when you calculate variance.
    So this is the same as V(aX + bY),
    since
    aX + b(Y + c) = aX + bY + bc
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  5. #5
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    Quote Originally Posted by matheagle View Post
    Her sigma_xy IS covariance in most books.
    So that's just notation.
    The only thing I see wrong is number 11.
    You're missing the covariance term.
    Instead of
    Calculate V(aX + b(Y + c)) = a^2sigmaX + b^2sigmaY
    you need to add 2ab times the covariance of X and Y.
    The constant term bc disappears when you calculate variance.
    So this is the same as V(aX + bY),
    since
    aX + b(Y + c) = aX + bY + bc
    But \sigma_X does stand for the standard deviation. And all the \sigma_X and \sigma_Y should be variances.
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  6. #6
    MHF Contributor matheagle's Avatar
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    Thats right..
    I didn't notice she left out the squares.
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