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Math Help - Correlation of Two Random Variables

  1. #1
    Junior Member atalay's Avatar
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    Correlation of Two Random Variables

    If X and Y have a bivariate normal distribution function with parameters μx, μy, σx̄, σy, ρ. Let U=X+Y and V=X-Y. Find an expression for correlation coefficient of U and V.

    I did this question but i am not sure because i am confused for U and V. How do this problem?
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  2. #2
    MHF Contributor matheagle's Avatar
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    Cov(X-Y,X+Y) =V(X)+Cov(X,Y)-Cov(X,Y)-V(Y) =V(X)-V(Y)

    V(X-Y) =V(X)-2Cov(X,Y)+V(Y)

    V(X+Y) =V(X)+2Cov(X,Y)+V(Y)

    So V(X+Y)V(X-Y) =\left(V(X)+V(Y)\right)^2-4\left(Cov(X,Y)\right)^2
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  3. #3
    Junior Member atalay's Avatar
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    Thanks, i did.

    i found that

    μu=μx+μy and μv=μx-μy σu=σv=σx^2+σy^2+2ρσxσx


    Corr(X,Y)= ρ, so Cov (X,Y)= ρσxσy


    Cov(U,V)=Cov(X,X)+(-1)Cov(X,Y)+Cov(Y,X)+(-1)Cov(Y,Y)=Var (X)-Var(Y)


    Corr(U,V)=Cov(U,V)/σuσv=σx^2-σy^2/σuσv
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  4. #4
    MHF Contributor matheagle's Avatar
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    this part is incorrect

    Quote Originally Posted by atalay View Post
    Thanks, i did.

    σu=σv=σx^2+σy^2+2ρσxσx
    one has a plus and the other a minus infront of the covariance term
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