# To find the covariance given the joint probability density function.

Question:

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