Uncorrelated random variables that are not independent

Counterexample

The claim "uncorrelated random variables are independent" is false. Here are two counterexamples. First: let $X \sim \mathcal{N}(0, 1)$ and $Y = X^2$. Then $\operatorname{Cov}(X, Y) = E[X^3] - E[X] \, E[X^2] = 0 - 0 = 0$, since the odd moments of $X$ are $0$. But $X$ and $Y$ are not independent: $P(|X| \le 1, \, Y > 1) = 0$, while $P(|X| \le 1) \, P(Y > 1) > 0$. Second: let $X \sim \mathcal{N}(0, 1)$, and let $W$ be independent of $X$ with $P(W = 1) = P(W = -1) = 1/2$. Put $Y = WX$. Then $Y \sim \mathcal{N}(0, 1)$ and $\operatorname{Cov}(X, Y) = E[W] \, E[X^2] = 0$. But $|Y| = |X|$, so $X$ and $Y$ are not independent.

Remarks

In the second example, both $X$ and $Y$ are normal, but the pair $(X, Y)$ is not multivariate normal: $X + Y$ is $0$ with probability $1/2$. So the rule "uncorrelated implies independent" needs a joint normal law, not only normal marginals. The direction that is true is that independent variables are uncorrelated.

Explanations

L1 human

Covariance only detects linear relations. In the first example, $Y$ is a function of $X$, so it depends on $X$ completely, but the relation is not linear. So zero correlation does not mean that two variables have nothing to do with each other.