For $X\sim\mathcal{N}(0,1)$, find measurable functions $g,h$ s.t. $g(X),h(X)$ are uncorrelated

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)) \$$

• You answer is correct. You can even use $g = h = const.$. – Hans Engler Jan 19 at 16:00
• For an example such that g(X) and h(X) are uncorrelated but not independent, try g(x)=x and h(x)=x^2. – Did Jan 19 at 16:22

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.

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