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Probability, Bivariate Normal Distribution

  1. Jun 23, 2009 #1
    1. The problem statement, all variables and given/known data
    Let the probability density function of X and Y be bivariate normal. For what values of a is the variance of aX+Y minimum?


    2. Relevant equations
    The answer in the book is -p(X,Y)(std dev of X/std dev of Y)


    3. The attempt at a solution
    I think the equation for Var(aX+Y) is,
    Var(aX+Y)=a^2Var(X)+Var(Y)+2aCov(X,Y), but I have no idea how to work this equation to equal the answer in the book.

    Any ideas would be much appreciated!!
    1. The problem statement, all variables and given/known data



    2. Relevant equations



    3. The attempt at a solution
     
  2. jcsd
  3. Jun 24, 2009 #2

    statdad

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    Homework Helper

    Yes

    [tex]
    Var(aX+Y) = a^2 Var(X) + Var(Y) + 2 a Cov(X,Y)
    [/tex]

    which, as a function of [tex] a [/tex], looks like a quadratic function. How about you? (There's a hint here :) )
     
  4. Jun 24, 2009 #3
    Thanks, Statdad!! Your hint was just the insight I needed.
     
  5. Jul 1, 2009 #4
    I think I spoke too soon, Statdad.

    I can see where the equation Var(aX+Y) is a quadratic equation, but I still can't factor the equation to obtain "-p(X,Y)(std dev of Y/std dev of X)" as one of the factors where -p(X,Y) is the negative correlation coefficient.

    I am wondering whether I should be working with a different equation.

    Any further assistance would be appreciated.
     
  6. Jul 1, 2009 #5

    EnumaElish

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    Science Advisor
    Homework Helper

    Remember p(X,Y) = Cov(X,Y)/sqrt(Var(X)Var(Y)).
     
  7. Jul 2, 2009 #6
    When I try to factor this equation using the formula we learned in high school, I get

    -2Cov(X,Y) +-radical(4Cov^2(X,Y)-4VarXVarY)divided by 2VarX.

    Since everything under the radical goes to zero, I am left with

    -2Cov(X,Y)/2VarX = Cov(X,Y)/VarX; this is not the answer I should be coming up with.
     
  8. Jul 5, 2009 #7
    I still need an answer to this problem, so if anyone knows what I'm doing wrong here, I would appreciate the help.
     
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