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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.)


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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.

The Wikipedia article on "Uncorrelated" cites this example and gives another, more explicit, discrete example with calculations.

As is mentioned in the Wikipedia article on Correlation,

"The Pearson correlation coefficient indicates the strength of a linear relationship between two variables, but its value generally does not completely characterize their relationship. In particular, if the conditional mean of Y given X, denoted E(Y|X), is not linear in X, the correlation coefficient will not fully determine the form of E(Y|X)."

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.

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