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Calculating covariance from variances

  1. Oct 23, 2013 #1
    1. The problem statement, all variables and given/known data

    Suppose that X1, X2, and X3
    are independent random variables with variances 3, 4, and 8, respectively.
    Let Y1 = 2X1 + 3X2, Y2 = X3 – X2, and Y3 = X1 + X2 + X3. (a) Using the general
    relationship
    Cov(W+X, Y+Z) = Cov(W,Y) + Cov(W, Z) + Cov(X, Y) + Cov(X, Z), find
    Cov(Yi, Yj) for all i, j.

    2. Relevant equations



    3. The attempt at a solution

    I can set up the Cov(Yi, Yj) for all i, j easily enough, but I do not understand how to calculate, say, 2Cov(X1, X3) just from the variances of X1 and X3. I know this is trivial, but any help would be greatly appreciated.
     
  2. jcsd
  3. Oct 23, 2013 #2

    Ray Vickson

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    What is the covariance of two independent random variables?
     
  4. Oct 23, 2013 #3
    Oh, boy...I seem to have missed the "independent" condition in the problem. It works out nicely that each combination seems to have a random variable with a covariance of itself somewhere in there. The rest all becomes 0s.

    Thanks!
     
  5. Oct 24, 2013 #4

    Ray Vickson

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    b
    Also: to compute the covariances you need to know the means of the X_i, which seem to not have been given. Were they given, and you just forgot to include them here?
     
  6. Oct 26, 2013 #5
    They were not given but, thankfully, because they are independent, the only covariances that are not 0 are those that are just a covariance with itself, i.e. the variance already given--usually multiplied by some constant.
     
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