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Math Help - expectation, variance, covariance

  1. #1
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    expectation, variance, covariance

    I'm stuck on this problem :

    X and Y is normal random var with mean(x) = 1 and mean(y) = 2 and variance(x)=1 and variance(y) = 4 and covariance(x,y)=1.

    U = X-Y
    V = X+Y

    find E(U),Var(U),Cov(U,V)
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  2. #2
    Super Member girdav's Avatar
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    Re: expectation, variance, covariance

    E(U) =E(X)-E(Y), \mathrm{Var}(U) = E((X-Y)^2)-(E(X-Y))^2 and \mathrm{Cov}(U,V)= E(UV)-E(U)E(V).
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  3. #3
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    Re: expectation, variance, covariance

    Quote Originally Posted by girdav View Post
    E(U) =E(X)-E(Y), \mathrm{Var}(U) = E((X-Y)^2)-(E(X-Y))^2 and \mathrm{Cov}(U,V)= E(UV)-E(U)E(V).
    I thought that above only if x and y independent ?
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  4. #4
    Super Member girdav's Avatar
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    Re: expectation, variance, covariance

    E(X_1+X_2)=E(X_1)+E(X_2) for two integrable random variables is always true, but \mathrm{Var}(X_1+X_2)=\mathrm{Var}(X_1)+\mathrm{Va  r}(X_2) is true if X_1 and X_2 have a variance and E(X_1X_2)=E(X_1)E(X_2). It's true when X_1 and X_2 are independent, but the last equality can be true even if X_1 and X_2 are not independent.
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