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  "body": "\n## Definition\n\nLet $\\mu$ and $\\nu$ be [[probability distributions|0QB913WM5E]] on $\\mathbb{R}$.\nTheir convolution is the probability distribution $\\mu * \\nu$ on $\\mathbb{R}$ with\n$$(\\mu * \\nu)(B) = \\int_{\\mathbb{R}} \\nu(B - x) \\, \\mu(dx)$$\nfor each Borel set $B \\subseteq \\mathbb{R}$, where $B - x = \\{b - x : b \\in B\\}$.\n\n## Remarks\n\nIf $X \\sim \\mu$ and $Y \\sim \\nu$ are [[independent|0E9502NT92]], then $X + Y \\sim \\mu * \\nu$.\nFor distributions on the integers with mass functions $p$ and $q$, $\\mu * \\nu$ has the mass function $(p * q)(n) = \\sum_{k} p(k) \\, q(n - k)$.\nIf $\\mu$ and $\\nu$ have densities $f$ and $g$, then $\\mu * \\nu$ has the density $(f * g)(x) = \\int f(y) \\, g(x - y) \\, dy$.\nThe convolution is commutative and associative. Its characteristic function is the product: $\\varphi_{\\mu * \\nu} = \\varphi_\\mu \\varphi_\\nu$.\n\n## Explanations\n\n### L1 human\n\nThe convolution answers the question: \"What is the law of $X + Y$, if $X$ and $Y$ are independent?\"\nFor two fair dice, the convolution of the two uniform laws on $\\{1, \\dots, 6\\}$ gives the familiar triangle-shaped law of the sum on $\\{2, \\dots, 12\\}$.\n",
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