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Let $X$ be a random variable. $f$ and $g$ are two non-trivial (e.g. non-constant) measurable functions defined on the range of $X$. Can $f(X)$ and $g(X)$ be independent? Thanks!

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Any constraints on the probability space on which $X$ is defined? For a uniformly distributed $X$ on a finite space with a composite number of elements this is possible. – WimC Jul 9 '13 at 1:34
I'm not sure they can be independent, but they can certainly be uncorrelated. – Omnomnomnom Jul 9 '13 at 1:47
up vote 1 down vote accepted

$f$ and $g$ can be independent if they are simple functions that attain finitely many values. For example: take these functions on $[0,4]$

$$f(x)= \begin{cases} \frac 12 & \lfloor x \rfloor \text{ is even}\\ 0 & \text{otherwise} \end{cases}\\ g(x)= \begin{cases} \frac12 & \lfloor x/2 \rfloor \text{ is even}\\ 0 & \text{otherwise} \end{cases} $$

We have $$P(g(x)=C|f(x)=D)=P(g(x) = C)$$ and $$P(f(x)=C|g(x)=D)=P(f(x) = C)$$

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