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Math Help - Combining normal random variable question

  1. #1
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    Combining normal random variable question

    I am stuck on this question...

    I know for part a) that E(X-Y) = -1 and Var(X-Y) = 5, but I'm not sure where to go from here...

    Is X-Y still a normal random variable? If so then I get Norm(-1,5) but I don't know where to go from there either.

    For part b) I don't understand what the correlation coefficient is and so I don't know how to go about doing this.

    All help will be much appreciated
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  2. #2
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    Quote Originally Posted by neild View Post
    I am stuck on this question...

    I know for part a) that E(X-Y) = -1 and Var(X-Y) = 5, but I'm not sure where to go from here...

    Is X-Y still a normal random variable? If so then I get Norm(-1,5) but I don't know where to go from there either.

    For part b) I don't understand what the correlation coefficient is and so I don't know how to go about doing this.

    All help will be much appreciated
    The linear combination of two independent normal variates is well known to be a normal variate. See Normal distribution - Wikipedia, the free encyclopedia
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  3. #3
    MHF Contributor matheagle's Avatar
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    W=X-Y\sim N(-1,5)

    So  P(|W|>3) =P(W<-3)+P(W>3)

     =P(Z< {-3+1\over\sqrt{5}}) +P(Z> {3+1\over\sqrt{5}})

     =P(Z< {-3+1\over\sqrt{5}}) +1-P(Z< {3+1\over\sqrt{5}})

     =1+\Phi({-2\over\sqrt{5}}) -\Phi({4\over\sqrt{5}})
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  4. #4
    MHF Contributor matheagle's Avatar
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    The correlation is the covariance divided by the two st deviations.
    I'll only do the covariance...

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

     = 2V(X)+0-0-V(Y)

    =-2, which is legal.
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