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  "body": "\n## Definition\n\nLet $x_0 \\in \\mathbb{R}$ and $\\gamma \u003e 0$.\nThe Cauchy distribution with location $x_0$ and scale $\\gamma$ is the distribution on $\\mathbb{R}$ with the [[probability density function|04YG690N5V]]\n$$f(x) = \\frac{1}{\\pi \\gamma \\left(1 + \\left(\\frac{x - x_0}{\\gamma}\\right)^2\\right)} = \\frac{\\gamma}{\\pi \\left((x - x_0)^2 + \\gamma^2\\right)}, \\quad x \\in \\mathbb{R}.$$\nThe standard Cauchy distribution has $x_0 = 0$ and $\\gamma = 1$, with density $\\frac{1}{\\pi(1 + x^2)}$.\n\n## Remarks\n\nThe cumulative distribution function is $F(x) = \\frac{1}{2} + \\frac{1}{\\pi} \\arctan\\left(\\frac{x - x_0}{\\gamma}\\right)$. The median is $x_0$.\nThe integral $\\int |x| f(x) \\, dx$ is infinite, so the distribution has no mean. It also has no variance.\nIf $X_1, \\dots, X_n$ are independent standard Cauchy, then their average $(X_1 + \\dots + X_n)/n$ is again standard Cauchy. So the law of large numbers fails without a finite mean.\nThe ratio $X/Y$ of two independent standard normal random variables is standard Cauchy.\n\n## Explanations\n\n### L1 human\n\nThe Cauchy distribution looks like a bell curve, but its tails are much heavier: very large values are not rare.\nShine a lamp at a uniform random angle onto a wall. The point where the light hits the wall has a Cauchy distribution.\n",
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