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Given Cont.Joint PDF function find Covariance MATRIX

  1. Jun 10, 2010 #1
    Hello Buddies,

    I need to calculate "covariance matrix" of the given joint PDF function.

    Joint PDF is fx(x1,x2,x3)=2/3(x1+x2+x3)

    over S (x1,x2,x3), 0<xi<1, i=1,2,3

    How can I calculte the Covariance Matrix?

  2. jcsd
  3. Jun 10, 2010 #2


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    Homework Helper

    There's a lot of symmetry to exploit here. Here is an outline of the brute force approach.

    Step 1: Find [tex] E(X_1) [/tex]. What will that tell you about the other 2 means.

    Step 2: Find [tex] E(X_1^2) [/tex], from this and Step 1 you can get [itex] \sigma_{X_1}^2 [/itex] (again - what about the other two?)

    Step 3: Now get the covariances between [itex] X_1 [/itex] and [itex] X_2 [/itex] and [itex] X_1 [/itex] and [itex] X_3 [/itex].

    Once you have these things you can assemble the covariance matrix.

    By the way: I'm assuming you meant that

    f(x_1, x_2, x_3) = \frac 2 3 \left(x_1 + x_2 + x_3\right), \quad 0 < x_1, x_2, x_3 < 1

    rather than something like

    \frac{2}{3(x_1 + x_2 + x_3)}, \quad 0<x_1, x_2, x_3 < 1

    since the second version isn't a density function.
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