0
$\begingroup$

Question: Provide an example of two random variables $X,Y$ and two functions $f,g :\mathbb{R} \rightarrow \mathbb{R}$ such that $X$ and $Y$ are not independent but $f(X)$ and $ g(Y)$ are independent. You may not pick $f$ and $g$ such that either $f(X)$ or $g(Y)$ is constant with probability one.

I got this question in Probablity Theory class and I've been trying to come up with an answer but everything I come up with turns out to be a constant function which is not allowed. Could anyone help me?

$\endgroup$

1 Answer 1

1
$\begingroup$

Let $X$ be a Rademacher random variable, i.e. $P(X = 1) = P(X = -1) = \dfrac{1}{2}$, and $N$ a non-constant positive random variable that is independent of $X$ (e.g. $N \sim U([1, 2])$). Now set $Y = XN$.

$X$ and $Y$ are clearly not independent, since $$P(Y > 0, X > 0) = P(X > 0) = \dfrac{1}{2} \ne \dfrac{1}{4} = P(Y > 0) P(X > 0)$$ holds. But $|Y| = N$ is independent of $X$.

$\endgroup$

You must log in to answer this question.