29:null 2a:["$","title",null,{"children":"Student-t distribution (standard) | Mattias Villani | Observable"}] 2f:[] 2d:{"id":"46614676a5e2184d","slug":"student-t-distribution-standard","title":"Student-t distribution (standard)","thumbnail":"86bccb87920720f8b58dc44bba3aeafbbb359ec496d2e7a3d0a2caf600cc7e8a","access_level":"public","create_time":"$D2025-05-05T11:53:59.759Z","update_time":"$D2025-05-05T12:00:07.870Z","view_time":null,"first_public_time":"$D2025-05-05T11:53:59.759Z","has_unpublished_edits":null,"fork":{"id":"4594bdcee9954fb1","slug":"student-t-distribution","title":"Student-t distribution (three-parameter)","owner":{"id":"7592a7ab121f74a0","login":"mattiasvillani","name":"Mattias Villani"}},"stars":0,"starred":false,"owner":{"type":"team","id":"7592a7ab121f74a0","login":"mattiasvillani","name":"Mattias Villani","avatar_url":"https://avatars-next.observableusercontent.com/a440aaf39a3abefb86cbcc758fd7a5ab42555ae68d6825b52ec006f385a94d1d","tier":"education_2026"},"version":558,"publish_version":null,"role":null,"authored":false,"authors":[{"type":"individual","id":"56499be8cf43f0ef","login":"mattiasvillani","name":"Mattias Villani","avatar_url":"https://avatars-next.observableusercontent.com/a440aaf39a3abefb86cbcc758fd7a5ab42555ae68d6825b52ec006f385a94d1d","tier":null}],"members":[{"type":"individual","id":"56499be8cf43f0ef","login":"mattiasvillani","name":"Mattias Villani","avatar_url":"https://avatars-next.observableusercontent.com/a440aaf39a3abefb86cbcc758fd7a5ab42555ae68d6825b52ec006f385a94d1d","tier":null,"role":"editor"},{"type":"team","id":"7592a7ab121f74a0","login":"mattiasvillani","name":"Mattias Villani","avatar_url":"https://avatars-next.observableusercontent.com/a440aaf39a3abefb86cbcc758fd7a5ab42555ae68d6825b52ec006f385a94d1d","tier":"education_2026","role":"editor"}],"comments":[],"custom_thumbnail":false,"trashed":false,"lapsed":null,"subscribed":null,"forks":0,"collection":{"id":"40c8ef5af1a51aff","slug":"distributions","title":"Statistical distributions","owner":{"id":"7592a7ab121f74a0","login":"mattiasvillani","name":"Mattias Villani"}},"type":"notebook","body":{"title":"Student-t distribution (standard)","cells":[{"id":38,"value":"# Student-*t* distribution (standard)","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","output":"","database":"$undefined"},{"id":523,"value":"

${tex`X \\sim t_{\\nu}`}, for ${tex`-\\infty \\leq X \\leq \\infty`}","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","output":"","database":"$undefined"},{"id":406,"value":"The density of a ${tex`X \\sim t_{\\nu}`} random variable is\n\n${tex`f(x) = \\frac{\\Gamma(\\frac{\\nu + 1}{2})}{\\Gamma(\\frac{\\nu}{2})\\sqrt{\\pi\\nu}\\,} \\left(1+\\frac{x^2}{\\nu} \\right)^{-(\\nu+1)/2}`}","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","output":"","database":"$undefined"},{"id":408,"value":"moments = tex`\n\\begin{aligned}\n&E( X) = 0 \\,\\,\\text{ if }\\nu > 1 \\\\[1em]\n&Var( X) = \\frac{\\nu}{\\nu-2} \\,\\,\\text{ if } \\nu >2 \n\\end{aligned}\n`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":118,"value":"viewof myinputs = Inputs.form([\n Inputs.range([1, 50], {value: 4, step: 1, label: tex`\\text{degrees of freedom }\\nu`}),\n Inputs.range([-10, 10], {value: -1.96, step: 0.01, label: \"quantile\"})\n ])","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":450,"value":"viewof shownormal = Inputs.toggle({value: false, label: \"show normal\"})","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":462,"value":"tailprobs = \n{\n var tailprobs;\n if (shownormal){\n tailprobs = tex`\\text{Normal distribution: } P(X\\leq ${myinputs[1]})=${normcdf.toPrecision(4)} \\\\ \\text{Student-}t \\text{ distribution: } P(X\\leq ${myinputs[1]})=${studentcdf.toPrecision(4)}`\n }\n else{\n tailprobs = tex`\\text{Student-}t \\text{ distribution: } P(X\\leq ${myinputs[1]})=${studentcdf.toPrecision(4)}`\n }\n return tailprobs;\n}","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":305,"value":"plt = Plot.plot({\n color: {\n legend: shownormal\n },\n x: {\n label: \"x\",\n axis: true\n },\n y: {\n axis: false,\n domain: [0,1.05*jstat.normal.pdf(0, 0, 1)]\n },\n marks: [\n Plot.ruleY([0]),\n Plot.line(data, {filter: d => d.dist == \"student-t\", x: \"x\", y: \"pdf\", stroke : \"dist\", strokeWidth: 2}),\n Plot.line(data, {filter: d => d.dist == \"normal\" && shownormal, x: \"x\", y: \"pdf\", stroke : \"dist\", strokeWidth: 2}),\n Plot.areaY(data, {filter: d => d.x <= myinputs[3] && d.dist == \"normal\" && shownormal, x: \"x\", y: \"pdf\", fill: \"steelblue\", opacity: 0.2}),\n Plot.areaY(data, {filter: d => d.x <= myinputs[3] && d.dist == \"student-t\", x: \"x\", y: \"pdf\", fill: \"orange\", opacity: 0.2}),\n \n ]\n })","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":515,"value":"## Relations to other distributions\n\nIf ${tex`X \\sim t_\\nu(0,1)`} follows a standard *t*-distribution, then ${tex`\\mu + \\tau X \\sim t_\\nu(\\mu,\\tau^2)`}","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","output":"","database":"$undefined"},{"id":109,"value":"
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\n\n---\n## 🙈 Appendix 🙈\n\nHere is only code and details that you probably don't want to read.","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","output":"","database":"$undefined"},{"id":221,"value":"#### Imports","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","output":"","database":"$undefined"},{"id":26,"value":"jstat = require('jstat')","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":223,"value":"#### Data","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","output":"","database":"$undefined"},{"id":17,"value":"data = {\n const x = d3.range(-10, 10, 0.01);\n const normpdf = x.map(x => ({x: x, pdf: jstat.normal.pdf(x, 0, 1), dist: \"normal\"}));\n const studentpdf = x.map(x => ({x: x, pdf: jstat.studentt.pdf((x-0), myinputs[0]), dist: \"student-t\"}));\n const data = normpdf.concat(studentpdf)\n return data\n}","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":19,"value":"normcdf = jstat.normal.cdf(myinputs[1],0,1) ","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":23,"value":"studentcdf = jstat.studentt.cdf(myinputs[1], myinputs[0]);","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":229,"value":"#### Style","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","output":"","database":"$undefined"},{"id":202,"value":"","pinned":false,"hidden":false,"mode":"html","author":"$undefined","summary":"$undefined","output":"","database":"$undefined"},{"id":58,"value":"viewof md_colour = Inputs.color({label: \"Highlight color cells\", value: \"#F5F5DC\"})","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"}],"files":"$W2f","stdlib":"1","titling":"auto"}} 2e:"$2d:body"