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  "body": "\n## Definition\n\nLet $\\mu \\in \\mathbb{R}$ and $\\sigma^2 \u003e 0$.\nThe density of the normal distribution with mean $\\mu$ and variance $\\sigma^2$ is the function $\\varphi_{\\mu, \\sigma^2} \\colon \\mathbb{R} \\to (0, \\infty)$ with\n$$\\varphi_{\\mu, \\sigma^2}(x) = \\frac{1}{\\sqrt{2\\pi\\sigma^2}} \\exp\\left(-\\frac{(x - \\mu)^2}{2\\sigma^2}\\right).$$\nThe standard normal density is $\\varphi(x) = \\varphi_{0, 1}(x) = \\frac{1}{\\sqrt{2\\pi}} e^{-x^2/2}$.\n\n## Remarks\n\nThe integral of $\\varphi_{\\mu, \\sigma^2}$ over $\\mathbb{R}$ is $1$. This follows from the Gaussian integral $\\int_{-\\infty}^{\\infty} e^{-x^2} \\, dx = \\sqrt{\\pi}$.\nSo $\\varphi_{\\mu, \\sigma^2}$ is a [[probability density function|04YG690N5V]], the density of the [[normal distribution|07RXBVTVZB]] $\\mathcal{N}(\\mu, \\sigma^2)$.\nThe function is symmetric about $\\mu$, has its maximum $1/\\sqrt{2\\pi\\sigma^2}$ at $x = \\mu$, and has inflection points at $\\mu \\pm \\sigma$.\nIt satisfies $\\varphi_{\\mu, \\sigma^2}(x) = \\frac{1}{\\sigma} \\varphi\\left(\\frac{x - \\mu}{\\sigma}\\right)$ with $\\sigma = \\sqrt{\\sigma^2}$.\n\n## Explanations\n\n### L1 human\n\nThis is the formula of the bell curve.\nThe exponent $-(x - \\mu)^2 / (2\\sigma^2)$ makes the curve fall very fast away from $\\mu$, and the constant in front makes the total area equal to $1$.\n",
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