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  "body": "\n## Definition\n\nLet $\\nu \u003e 0$ be a real number.\nStudent's t-distribution with $\\nu$ degrees of freedom, written $t_\\nu$, is the law on $\\mathbb{R}$ with the density\n$$f(t) = \\frac{\\Gamma\\left(\\frac{\\nu + 1}{2}\\right)}{\\sqrt{\\nu \\pi}\\, \\Gamma\\left(\\frac{\\nu}{2}\\right)} \\left(1 + \\frac{t^2}{\\nu}\\right)^{-(\\nu + 1)/2}, \\qquad t \\in \\mathbb{R}.$$\n\n## Remarks\n\nLet $Z$ be standard normal and $V \\sim \\chi^2_\\nu$ (see [[chi-squared distribution|0RFMMBHFTV]]), [[independent|0XCE070V8B]]. Then $Z / \\sqrt{V / \\nu} \\sim t_\\nu$.\nFor an i.i.d. $\\mathcal{N}(\\mu, \\sigma^2)$ sample of size $n \\ge 2$, $\\frac{\\bar{X}_n - \\mu}{S_n / \\sqrt{n}} \\sim t_{n - 1}$, with the [[sample mean|0QNRBHWV35]] $\\bar{X}_n$ and the [[sample standard deviation|0JN9P4A944]] $S_n$.\nThe mean is $0$ for $\\nu \u003e 1$, and the variance is $\\nu / (\\nu - 2)$ for $\\nu \u003e 2$. For $\\nu = 1$, $t_1$ is the standard [[Cauchy distribution|0NA6Q82YR7]].\nAs $\\nu \\to \\infty$, the density of $t_\\nu$ converges to the standard normal density.\n\n\n\n## Explanations\n\n### L1 human\n\nThe t-distribution looks like the bell curve, but it gives more probability to values far from $0$.\nIt takes into account the extra uncertainty when the standard deviation comes from a small sample. With many degrees of freedom, it is almost the normal law.\n",
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