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  "body": "\n## Definition\n\nLet $\\mu$ be a probability distribution on $\\mathbb{R}^2$, for example the [[joint distribution|0QAR0XDZH4]] of real random variables $X$ and $Y$.\nThe first marginal distribution of $\\mu$ is the distribution $\\mu_1$ on $\\mathbb{R}$ with $\\mu_1(A) = \\mu(A \\times \\mathbb{R})$ for each Borel set $A \\subseteq \\mathbb{R}$. The second marginal is $\\mu_2(B) = \\mu(\\mathbb{R} \\times B)$.\nIf $\\mu$ is the law of $(X, Y)$, then $\\mu_1$ is the law of $X$ and $\\mu_2$ is the law of $Y$.\n\n## Remarks\n\nThe same definition works on a product $E_1 \\times E_2$ of measurable spaces, and for $n$ variables with the projections onto some of the coordinates.\nIn the discrete case, $P(X = x) = \\sum_{y} P(X = x, Y = y)$. In the case of a joint density $f(x, y)$, $X$ has the density $f_X(x) = \\int_{-\\infty}^{\\infty} f(x, y) \\, dy$.\nThe name comes from tables: the sums of the rows and the columns of a joint table are written in its margins.\n\n## Explanations\n\n### L1 human\n\nTo get the marginal distribution of $X$ from a joint table of $X$ and $Y$, add up each row.\nYou lose the information about how $X$ and $Y$ are linked, and you keep only the chances of the values of $X$.\n",
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