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  "body": "\n## Statement\n\nLet $X$ and $Y$ be random variables on one probability space, with values in measurable spaces $(E_1, \\mathcal{E}_1)$ and $(E_2, \\mathcal{E}_2)$.\nLet $\\mu_X$ and $\\mu_Y$ be their laws, and let $\\mu_{(X, Y)}$ be the [[joint law|0QAR0XDZH4]] of the pair $(X, Y)$ on $E_1 \\times E_2$.\nThen $X$ and $Y$ are [[independent|0E9502NT92]] if and only if $\\mu_{(X, Y)} = \\mu_X \\otimes \\mu_Y$, the product measure of the two [[marginal laws|0AXP2Z5N9H]].\n\n## Remarks\n\nThe statement is a direct translation of the definition: $\\mu_{(X, Y)}(A \\times B) = P(X \\in A, Y \\in B)$ and $(\\mu_X \\otimes \\mu_Y)(A \\times B) = \\mu_X(A) \\, \\mu_Y(B)$.\nTwo probability measures on $E_1 \\times E_2$ that agree on all rectangles $A \\times B$ are equal, because the rectangles form a π-system that generates the product σ-algebra.\nFor real random variables, it follows that $F_{(X, Y)}(x, y) = F_X(x) \\, F_Y(y)$. If $X$ and $Y$ have densities, the joint density is $f_X(x) \\, f_Y(y)$.\nThe same result holds for $n$ random variables: they are independent if and only if their joint law is $\\mu_{X_1} \\otimes \\dots \\otimes \\mu_{X_n}$.\n\n## Explanations\n\n### L1 human\n\nFor independent variables, the full joint picture is fixed by the two separate pictures.\nYou find the chance of a pair of outcomes by multiplying the chances of the two outcomes.\n",
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