2b:null 2c:["$","title",null,{"children":"Chi2-distribution | Mattias Villani | Observable"}] 31:[] 2f:{"id":"67961e538501b542","slug":"chi2-distribution","title":"Chi2-distribution","thumbnail":"f6e3c916f9c5474731ee62e51cb4e350ac33e5e81282f597cd348fd93a826bcd","access_level":"public","create_time":"$D2023-02-25T19:18:44.604Z","update_time":"$D2024-02-18T18:07:13.069Z","view_time":null,"first_public_time":"$D2023-02-25T19:18:44.604Z","fork":{"id":"1b150f86eb91a63a","slug":"chi2-fordelningen","title":"Chi2-fördelningen","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":657,"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":3,"collection":{"id":"40c8ef5af1a51aff","slug":"distributions","title":"Statistical distributions","owner":{"id":"7592a7ab121f74a0","login":"mattiasvillani","name":"Mattias Villani"}},"type":"notebook","body":{"title":"Chi2-distribution","cells":[{"id":38,"value":"# Chi2-distribution","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined"},{"id":629,"value":"

${tex`X\\sim \\chi^2(\\nu)`}, for ${tex`X>0`}","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined"},{"id":496,"value":"The density function of a ${tex`X\\sim \\chi^2(\\nu)`} random variable is\n\n${tex`f(x)=\\frac{1}{2^{\\nu/2}\\Gamma(\\nu/2)}\\; x^{\\nu/2-1} e^{-x/2}`}","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined"},{"id":118,"value":"viewof params = Inputs.form([\n Inputs.range([1, 20], {value: 4, step: 1, label: tex`\\nu`}),\n Inputs.range([0.01, 10], {value: 2, step: 0.01, label: \"quantile:\"})\n ])","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":461,"value":"dist_quantile = tex`\\text{If } X \\sim \\chi^2(${params[0]}) \\text{ then } `","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":555,"value":"moments = tex`\n\\begin{aligned}\n&E( X) =\\nu = ${params[0].toPrecision(2)} \\\\[1em]\n&Var( X) =2\\nu= ${(2*params[0]).toPrecision(2)} \\\\[1em]\n&P( X\\leq ${params[1]}) =${chisquarecdf.toPrecision(4)}\n\\end{aligned}\n`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":305,"value":"plt_pdf = Plot.plot({\n color: {\n legend: true\n },\n x: {\n label: \"x\",\n axis: true\n },\n y: {\n label: \"f(x)\",\n //axis: false,\n domain: [0,jstat.chisquare.pdf(Math.max(0,params[0]-2), params[0])]\n },\n marks: [\n Plot.ruleY([0]),\n Plot.ruleX([0]),\n Plot.line(chisquarepdf, {x: \"x\", y: \"pdf\", stroke : \"steelblue\", strokeWidth: 2}),\n Plot.areaY(chisquarepdf, {filter: d => d.x <= params[1], x: \"x\", y: \"pdf\", fill: \"steelblue\", opacity: 0.2})\n ]\n })","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":610,"value":"## Relations to other distributions","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined"},{"id":612,"value":">

If ${tex`X \\sim \\operatorname{N}(0,1)`} then\n>

${tex`X^2 \\sim \\chi^2(1)`}

\n\n> that is, the square of a standard normally distributed variable is ${tex`\\chi^2`}-distributed with one degree of freedom ${tex`\\nu=1`}.\n ","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined"},{"id":625,"value":"Summing ${tex`\\nu`} squared independent standard normal variables gives a ${tex`\\chi^2`} variable with ${tex`\\nu`} degrees of freedom:","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined"},{"id":617,"value":">

If ${tex`X_1,X_2,\\ldots,X_{\\nu} \\overset{\\mathrm{iid}}{\\sim} \\operatorname{N}(0,1)`} then\n>

${tex`\\sum_{i=1}^{\\nu}X_i^2 \\sim \\chi^2(\\nu)`}

\n ","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined"},{"id":109,"value":"
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\n\n---\n## 🙈 Appendix 🙈\n\nHär följer bara kod och detaljer som du antagligen inte vill läsa.","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined"},{"id":221,"value":"#### Imports","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined"},{"id":26,"value":"jstat = require('jstat')","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":223,"value":"#### Data","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined"},{"id":17,"value":"chisquarepdf = {\n const x = d3.range(0.01, params[0]+5*Math.sqrt(2*params[0]), 0.01);\n const pdf = x.map(x => ({x: x, pdf: jstat.chisquare.pdf(x, params[0])}));\n return pdf\n}","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":19,"value":"chisquarecdf = jstat.chisquare.cdf(params[1], params[0]);","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":229,"value":"#### Style","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined"},{"id":202,"value":"","pinned":false,"hidden":false,"mode":"html","author":"$undefined","summary":"$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"}],"files":"$W31","stdlib":"1","titling":"auto"}} 30:"$2f:body"