= Uncorrelated random variables
Two <square-integrable> <random variables> $X,Y$ are <uncorrelated random variables> when $\operatorname{Cov}(X,Y)=0$. <Independent random variables> with finite <second moments> are <uncorrelated random variables>, but zero <covariance> does not generally imply <independence>.
For an explicit counterexample, let $X$ take $-1$, $0$, $1$ with equal <probabilities>, and let $Y=X^2$. Then $\mathbb E X=\mathbb E X^3=0$, so $\operatorname{Cov}(X,Y)=\mathbb E X^3-\mathbb E X\,\mathbb E X^2=0$. Nevertheless, $Y$ is completely determined by $X$: $\mathbb P(Y=0\mid X=0)=1$, while $\mathbb P(Y=0)=1/3$. Thus these are <uncorrelated random variables> with <statistical dependence>. In this example, opposite contributions from the <odd function> $x^3$ cancel; the <even function> $x^2$ still retains information about $|X|$. The last panel of the <covariance> diagram shows the relationship.
There are useful special cases. <Uncorrelated jointly normal variables are independent>, and the <Bernoulli independence from zero covariance> result supplies another setting where the converse holds. Their distribution assumptions do the extra work; zero <covariance> alone does not.
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