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  "body": "\n## Counterexample\n\nThe claim \"uncorrelated random variables are independent\" is false. Here are two counterexamples.\nFirst: 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 $0$.\nBut $X$ and $Y$ are not independent: $P(|X| \\le 1, \\, Y \u003e 1) = 0$, while $P(|X| \\le 1) \\, P(Y \u003e 1) \u003e 0$.\nSecond: let $X \\sim \\mathcal{N}(0, 1)$, and let $W$ be independent of $X$ with $P(W = 1) = P(W = -1) = 1/2$. Put $Y = WX$.\nThen $Y \\sim \\mathcal{N}(0, 1)$ and $\\operatorname{Cov}(X, Y) = E[W] \\, E[X^2] = 0$. But $|Y| = |X|$, so $X$ and $Y$ are not independent.\n\n## Remarks\n\nIn the second example, both $X$ and $Y$ are normal, but the pair $(X, Y)$ is not multivariate normal: $X + Y$ is $0$ with probability $1/2$.\nSo the rule \"uncorrelated implies independent\" needs a joint normal law, not only normal marginals.\nThe direction that is true is that [[independent variables are uncorrelated|077GN9HDR6]].\n\n## Explanations\n\n### L1 human\n\nCovariance 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.\nSo zero correlation does not mean that two variables have nothing to do with each other.\n",
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        "locator": "A symmetric example",
        "url": "https://en.wikipedia.org/wiki/Misconceptions_about_the_normal_distribution"
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        "locator": "Examples of dependence without correlation",
        "url": "https://en.wikipedia.org/wiki/Uncorrelatedness"
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  "msc": [
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  "name": "Uncorrelated random variables that are not independent",
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