Suppose the pdf of $(X_1,X_2,X_3,X_4)$ is given by $$f(x_1,x_2,x_3,x_4)=\begin{cases}\text{constant}&,\text{ if }x_1^2+x_2^2+x_3^2+x_4^2\le 1 \\ 0&,\text{ otherwise}\end{cases}$$

Find the covariance between $X_1$ and $X_4$. Are $X_1$ and $X_4$ independent? Justify your answer.

I was solving some question papers and got stuck in this problem.

My problem:

I know how to find "marginal probabilities" from a joint probability density function and also know how to find the covariance and independence.

But I am getting stuck in the range part while integration. While finding f(x1, x4) or f(x1) or during finding the covariance how do I integrate over the given range as I am not accustomed to handle such ranges as in here.(marked in red)

enter image description here

Please help me solve this.

Thank you.

  • 1
    $\begingroup$ First attempt the problem with two variables instead of four, so you can draw a picture of the range of integration. Look at it closely to see whether there are any symmetries you can exploit to avoid computing integrals altogether. $\endgroup$ – whuber Nov 25 '18 at 16:12
  • $\begingroup$ You could try transforming to spherical coordinates for the integrals. $\endgroup$ – StubbornAtom Nov 25 '18 at 16:13
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    $\begingroup$ @Stubborn Although that works, it is by far the most difficult way to solve this problem. $\endgroup$ – whuber Nov 25 '18 at 16:13
  • $\begingroup$ @whuber I thought of drawing it graphically but I somehow cpuldnot make it..would it be a circle as x1^2+x4^2《 1-x2^2-x3^2 , here after drawing the circle how can I deal with the inequality ? Its not clear..Can you please help me? $\endgroup$ – P db Nov 25 '18 at 16:20
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    $\begingroup$ With two variables it's not a circle; it's a disk defined by $x_1^2+x_2^2\le1.$ The distribution is uniform on this disk. Thus, merely by looking at the diagram, you can read off the conditional distributions $X_2\mid X_1$ and you can see some obvious symmetries in the joint distribution of $(X_1,X_2),$ such as the reflections with respect to the coordinate axes. Without any further consideration, that answers both parts of the question. Now generalize to more variables. See stats.stackexchange.com/questions/257859 or stats.stackexchange.com/questions/135663 for examples. $\endgroup$ – whuber Nov 25 '18 at 19:25