2a:null 2b:["$","title",null,{"children":"Student-t distribution (three-parameter) | Mattias Villani | Observable"}] 31:[] 2e:{"id":"4594bdcee9954fb1","slug":"student-t-distribution","title":"Student-t distribution (three-parameter)","thumbnail":"7fa9869fc67d54ade65ee726d9da32fe31157ec06c4f344b93826f6e35678371","access_level":"public","create_time":"$D2023-02-10T19:04:43.459Z","update_time":"$D2024-02-03T12:34:48.211Z","view_time":null,"first_public_time":"$D2023-02-10T19:04:43.459Z","fork":{"id":"e3dec291bc5401c5","slug":"student-t-fordelning-allman-version","title":"Student-t fördelning (allmän version)","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":"pro_2024"},"version":536,"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"}],"comments":[],"custom_thumbnail":false,"trashed":false,"subscribed":null,"forks":7,"collection":{"id":"40c8ef5af1a51aff","slug":"distributions","title":"Statistical distributions","owner":{"id":"7592a7ab121f74a0","login":"mattiasvillani","name":"Mattias Villani"}},"type":"notebook","body":{"title":"Student-t distribution (three-parameter)","cells":[{"id":38,"value":"# Student-*t* distribution (three-parameter)","pinned":false,"hidden":false,"mode":"md"},{"id":523,"value":"

${tex`X \\sim t_{\\nu}(\\mu,\\tau^2)`}, for ${tex`-\\infty \\leq X \\leq \\infty`}","pinned":false,"hidden":false,"mode":"md"},{"id":406,"value":"The density of a ${tex`X \\sim t_{\\nu}(\\mu,\\tau^2)`} random variable is\n\n${tex`f(x) = \\frac{\\Gamma(\\frac{\\nu + 1}{2})}{\\Gamma(\\frac{\\nu}{2})\\sqrt{\\pi\\nu}\\tau\\,} \\left(1+\\frac{1}{\\nu} \\left( \\frac{ x-\\mu } {\\tau } \\right)^2\\right)^{-(\\nu+1)/2}`}","pinned":false,"hidden":false,"mode":"md"},{"id":408,"value":"moments = tex`\n\\begin{aligned}\n&E( X) = \\mu \\,\\,\\text{ if }\\nu > 1 \\\\[1em]\n&Var( X) = \\frac{\\nu}{\\nu-2}\\tau^2 \\,\\,\\text{ if } \\nu >2 \n\\end{aligned}\n`","pinned":false,"hidden":false,"mode":"ojs"},{"id":118,"value":"viewof myinputs = Inputs.form([\n Inputs.range([-5, 5], {value: 0, step: 0.1, label: tex`\\text{location }\\mu`}),\n Inputs.range([0.1, 5], {value: 1, step: 0.1, label: tex`\\text{scale }\\tau`}),\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"},{"id":450,"value":"viewof shownormal = Inputs.toggle({value: false, label: \"show normal\"})","pinned":false,"hidden":false,"mode":"ojs"},{"id":462,"value":"tailprobs = \n{\n var tailprobs;\n if (shownormal){\n tailprobs = tex`\\text{Normal distribution: } P(X\\leq ${myinputs[3]})=${normcdf.toPrecision(4)} \\\\ \\text{Student-}t \\text{ distribution: } P(X\\leq ${myinputs[3]})=${studentcdf.toPrecision(4)}`\n }\n else{\n tailprobs = tex`\\text{Student-}t \\text{ distribution: } P(X\\leq ${myinputs[3]})=${studentcdf.toPrecision(4)}`\n }\n return tailprobs;\n}","pinned":false,"hidden":false,"mode":"ojs"},{"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(myinputs[0], myinputs[0], myinputs[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"},{"id":515,"value":"## Relations to other distributions\n\nIf ${tex`X \\sim t_\\nu(0,1)`} follows a standard *t*-distribution, then ${tex`\\mu + \\tau X \\sim t_\\nu(\\mu,\\tau^2)`}","pinned":false,"hidden":false,"mode":"md"},{"id":109,"value":"
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\n\n---\n## 🙈 Appendix 🙈\n\nHere is only code and details that you probably don't want to read.","pinned":false,"hidden":false,"mode":"md"},{"id":221,"value":"#### Imports","pinned":false,"hidden":false,"mode":"md"},{"id":26,"value":"jstat = require('jstat')","pinned":false,"hidden":false,"mode":"ojs"},{"id":223,"value":"#### Data","pinned":false,"hidden":false,"mode":"md"},{"id":17,"value":"data = {\n const x = d3.range(-10, 10, 0.01);\n const normpdf = x.map(x => ({x: x, pdf: jstat.normal.pdf(x, myinputs[0], myinputs[1]), dist: \"normal\"}));\n const studentpdf = x.map(x => ({x: x, pdf: jstat.studentt.pdf((x-myinputs[0])/myinputs[1], myinputs[2])/myinputs[1], dist: \"student-t\"}));\n const data = normpdf.concat(studentpdf)\n return data\n}","pinned":false,"hidden":false,"mode":"ojs"},{"id":19,"value":"normcdf = jstat.normal.cdf(myinputs[3], myinputs[0], myinputs[1]);","pinned":false,"hidden":false,"mode":"ojs"},{"id":23,"value":"studentcdf = jstat.studentt.cdf((myinputs[3]-myinputs[0])/myinputs[1], myinputs[2]);","pinned":false,"hidden":false,"mode":"ojs"},{"id":229,"value":"#### Style","pinned":false,"hidden":false,"mode":"md"},{"id":202,"value":"","pinned":false,"hidden":false,"mode":"html"},{"id":58,"value":"viewof md_colour = Inputs.color({label: \"Highlight color cells\", value: \"#F5F5DC\"})","pinned":false,"hidden":false,"mode":"ojs"}],"files":"$W31","stdlib":"1","titling":"auto"}} 2f:"$2e:body"