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  "body": "\n## Definition\n\nLet $\\mu \\in \\mathbb{R}$ and $\\sigma^2 \u003e 0$.\nThe normal distribution with mean $\\mu$ and variance $\\sigma^2$, written $\\mathcal{N}(\\mu, \\sigma^2)$, is the distribution on $\\mathbb{R}$ with the [[probability density function|04YG690N5V]]\n$$\\varphi_{\\mu, \\sigma^2}(x) = \\frac{1}{\\sqrt{2\\pi\\sigma^2}} \\exp\\left(-\\frac{(x - \\mu)^2}{2\\sigma^2}\\right), \\quad x \\in \\mathbb{R}.$$\nThe standard normal distribution is $\\mathcal{N}(0, 1)$.\n\n## Remarks\n\nThe function $\\varphi_{\\mu, \\sigma^2}$ is the [[density of the normal distribution|027JF1FBB0]].\nSome authors, and the library Mathlib, also allow $\\sigma^2 = 0$. Then $\\mathcal{N}(\\mu, 0)$ is the [[Dirac measure|099SXAG014]] $\\delta_\\mu$.\nIf $X \\sim \\mathcal{N}(\\mu, \\sigma^2)$ and $a \\ne 0$, $b$ are real, then $aX + b \\sim \\mathcal{N}(a\\mu + b, a^2 \\sigma^2)$. In particular, $(X - \\mu)/\\sigma \\sim \\mathcal{N}(0, 1)$.\nThe cumulative distribution function of $\\mathcal{N}(0, 1)$ is written $\\Phi$. It has no formula with elementary functions.\n\n## Explanations\n\n### L1 human\n\nThe normal distribution is the famous bell curve. It is symmetric about the mean $\\mu$, and $\\sigma$ measures its width.\nHeights of people and errors of measurement are often close to normal.\n",
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