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

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

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.