Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

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


    User Avatar
    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.
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook