Why the 3 random variables x, y, and z are independent??
Consider a random vector R which has coordinates x, y, and z (3 random variables).
Prove that the random vector R will be isotropically distributed in the 3D space only if the 3 coordinates are independent and identically distributed random variables.
This case is generally referred to as i.i.d variables (Independent and Identically Distributed).
This seems like someone's homework problem, so I'll just give a hint. Consider that the density function is isotropic about a mean that we can take as zero without loss of generality. Therefore, the density can be expressed as a function of radius only, and therefore a function of x^2+y^2+z^2. Note that the density is an "even" function, symmetric about zero under x, y, and z. Now consider any two variables, like x and y. How do you express the expectation value of the product (xy)? What do you know about even and odd functions that tells you something about this expectation? I'll stop there.
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