# To find the covariance given the joint probability density function [duplicate]

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)

• 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. – whuber Nov 25 '18 at 19:25