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Let $X\sim\mathcal{N}(0,1)$ a random variable. Find measurable function $g,h:\mathbb{R}\to\mathbb{R}$ such that $g(X),h(X)$ are uncorrelated.

I would like to confirm my answer:

Let $g,h:\mathbb{R}\to\mathbb{R}$ defined by $g(t)=1,h(t)=t$. $g$ and $h$ are measurable because ther're continuous. $$ \\ \mathbb{E}(g(X)h(X))=\mathbb{E}(h(X))=\mathbb{E}(h(x))\cdot1=\mathbb{E}(h(X))\cdot\mathbb{E}(g(X)) \ $$

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    $\begingroup$ You answer is correct. You can even use $g = h = const.$. $\endgroup$ – Hans Engler Jan 19 at 16:00
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    $\begingroup$ For an example such that g(X) and h(X) are uncorrelated but not independent, try g(x)=x and h(x)=x^2. $\endgroup$ – Did Jan 19 at 16:22
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Your answer is correct, but rather trivial. My guess is that whoever posed the question was hoping for a more interesting example than that one.

Another one to think about might be $g(x) = |x|$, $h(x) = \operatorname{sign} x$. And you might like to prove that they are not only uncorrelated but actually independent.

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  • $\begingroup$ My guess is that the purpose is the opposite: find g(X) and h(X) uncorrelated but not independent. $\endgroup$ – Did Jan 19 at 16:21

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