# In search of memorable example of “(Pearson-)uncorrelated $\not\Rightarrow$ independent”

I am looking for an easy-to-remember (and non-trivial) example that vividly illustrates that the "uncorrelatedness" (in the sense of Pearson) of two random variables $X, Y$ does not imply that $X$ and $Y$ are independent. By "non-trivial" I mean that all the joint probabilities are positive (whenever the associated marginals are). I realize that it may be too tall an order to come up with a non-trivial example that is sufficiently vivid or easy-to-remember, in which case, the non-triviality condition may be relaxed.

(Whether the example features a discrete or a continuous distribution is not important per se; what matters is that the example be simple enough to think through, ideally in one's head.)

Thanks!

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The standard example for this is considering a continuous, uniformly distributed variable $X$ on $[-1,1]$ and then looking at $X$ and $X^{2}$. They are uncorrelated, but obviously dependent.
In the particular case of the $X$, $X^{2}$ example I mentioned, we have that $E[X^{2}|X=x] = x^{2}$, which is not linear, and a linear approximation does a pretty terrible job away from $x=0$. That's the "why" behind it... we concocted a special random variable such that its conditional mean depends on $X$ in a precisely non-linear way. Since the Pearson coefficient is meant to measure the linearity of that conditional mean, it summarizes the relationship between this particular $X$ and $X^{2}$ as $0$... i.e. uncorrelated.