Independence and Conditional Independence between random variables

Question

For a family of random variables, I was wondering if independence and Conditional Independence under any condition among them imply each other?

If not, can these two concepts imply one another under some special cases? Thanks!

Answer 1

Independence does not imply conditional independence: for instance, independent random variables are rarely independent conditionally on their sum or on their maximum.

Conditional independence does not imply independence: for instance, conditionally independent random variables uniform on $(0,u)$ where $u$ is uniform on $(0,1)$ are not independent.

Answer 2

For the first question (independence does not imply conditional independence), one of my favorite simple counterexamples works. Flip two fair coins. Let $A$ be the event that the first coin is heads, $B$ the event that the second coin is heads, $C$ the event that the two coins are the same (both heads or both tails). Clearly $A$ and $B$ are independent, but they are not conditionally independent given $C$. If you want an example with random variables, consider the indicators $1_A, 1_B, 1_C$.

Of interest here is that $A,B,C$ are pairwise independent but not mutually independent (since any two determine the third).