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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!

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The question as you have stated it is vague. Is there some problem or result that is motivating this question? –  svenkatr Feb 16 '11 at 22:31
    
@svenkatr: Is it correct that independent random variables may not be independent conditional on some conditions? Independence conditional on some conditions may not imply independence. So I was wondering: (1) if Independence conditional on any condition and independence may imply each other; (2) what are some useful cases where one can imply the other? –  Tim Feb 16 '11 at 23:25
    
What is independence under any condition? If you mean independence under every possible conditioning, the condition is rarely fulfilled... For example, random variables $x$ and $y$ are not independent conditionally on $[x<y]$ if you exclude some very degenerate cases. –  Did Feb 17 '11 at 11:34

2 Answers 2

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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.

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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).

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