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  "body": "\n## Definition\n\nLet $X$ and $Y$ be real random variables on one probability space with $E[X^2] \u003c \\infty$ and $E[Y^2] \u003c \\infty$.\nThe covariance of $X$ and $Y$ is\n$$\\operatorname{Cov}(X, Y) = E\\left[(X - E[X]) \\, (Y - E[Y])\\right].$$\n$X$ and $Y$ are uncorrelated if $\\operatorname{Cov}(X, Y) = 0$.\n\n## Remarks\n\nAn equivalent formula is $\\operatorname{Cov}(X, Y) = E[XY] - E[X] \\, E[Y]$.\nThe covariance of $X$ with itself is the [[variance|007QED3E0Y]]: $\\operatorname{Cov}(X, X) = \\operatorname{Var}(X)$.\nThe covariance is symmetric and bilinear: $\\operatorname{Cov}(aX + bY, Z) = a \\operatorname{Cov}(X, Z) + b \\operatorname{Cov}(Y, Z)$. It does not change when one adds constants to $X$ or $Y$.\nThe Cauchy–Schwarz inequality gives $|\\operatorname{Cov}(X, Y)| \\le \\sigma_X \\sigma_Y$.\n\n## Explanations\n\n### L1 human\n\nCovariance says whether two quantities tend to move in the same direction.\nHeight and weight of people have positive covariance. The size of a covariance depends on the units, so one often uses the correlation coefficient to compare.\n",
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