Given Cont.Joint PDF function find Covariance MATRIX

[circuitman]

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?

Thanks

 

[statdad]

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

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

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

Step 3: Now get the covariances between X_1 and X_2 and X_1 and X_3.

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

By the way: I'm assuming you meant that

<br /> f(x_1, x_2, x_3) = \frac 2 3 \left(x_1 + x_2 + x_3\right), \quad 0 &lt; x_1, x_2, x_3 &lt; 1<br />

rather than something like

<br /> \frac{2}{3(x_1 + x_2 + x_3)}, \quad 0&lt;x_1, x_2, x_3 &lt; 1<br />

since the second version isn't a density function.