2d:null 2e:["$","title",null,{"children":"F-distribution | Mattias Villani | Observable"}] 33:T407,moments = { if (nu2>4){ return tex` \begin{aligned} &E( X) =\frac{\nu_2}{\nu_2-2} = ${(nu2/(nu2-2)).toPrecision(3)} \text{ if } \nu_2>2\\[1em] &Var( X) = 2\Big(\frac{\nu_2}{\nu_2-2}\Big)^2\frac{\nu_1+\nu_2-2}{\nu_1(\nu_2-4)} = ${(2*(nu2/(nu2-2))**2 * ((nu1+nu2-2)/(nu1*(nu2-4)))).toPrecision(3)} \text{ if } \nu_2>4 \\[1em] &P( X\leq ${quantile}) =${fcdf.toPrecision(4)} \end{aligned} ` } else{ if (nu2>2 && nu2<=4){ return tex` \begin{aligned} &E( X) =\frac{\nu_2}{\nu_2-2} = ${(nu2/(nu2-2)).toPrecision(3)} \text{ if } \nu_2>2\\[1em] &Var( X) = \text{ does not exist if } \nu_2>4 \\[1em] &P( X\leq ${quantile}) =${fcdf.toPrecision(4)} \end{aligned} ` } else{ return tex` \begin{aligned} &E( X) =\text{ does not exist if } \nu_2>2\\[1em] &Var( X) = \text{ does not exist if } \nu_2>4 \\[1em] &P( X\leq ${quantile}) =${fcdf.toPrecision(4)} \end{aligned} ` } } }34:[] 31:{"id":"372affbdf9c3bc62","slug":"f-distribution","title":"F-distribution","thumbnail":"224b498014522e2431b31aa728115c8632ad1d386109f2fbf4a020b4f1dd3a7a","access_level":"public","create_time":"$D2023-04-24T09:19:36.832Z","update_time":"$D2023-09-04T09:18:00.626Z","view_time":null,"first_public_time":"$D2023-04-24T09:19:36.832Z","fork":{"id":"67961e538501b542","slug":"chi2-distribution","title":"Chi2-distribution","owner":{"id":"7592a7ab121f74a0","login":"mattiasvillani","name":"Mattias Villani"}},"stars":1,"starred":false,"owner":{"type":"team","id":"7592a7ab121f74a0","login":"mattiasvillani","name":"Mattias Villani","avatar_url":"https://avatars-next.observableusercontent.com/a440aaf39a3abefb86cbcc758fd7a5ab42555ae68d6825b52ec006f385a94d1d","tier":"pro_2024"},"version":694,"publish_version":null,"role":null,"authored":false,"authors":[{"type":"individual","id":"56499be8cf43f0ef","login":"mattiasvillani","name":"Mattias Villani","avatar_url":"https://avatars-next.observableusercontent.com/a440aaf39a3abefb86cbcc758fd7a5ab42555ae68d6825b52ec006f385a94d1d","tier":null}],"members":[{"type":"individual","id":"56499be8cf43f0ef","login":"mattiasvillani","name":"Mattias Villani","avatar_url":"https://avatars-next.observableusercontent.com/a440aaf39a3abefb86cbcc758fd7a5ab42555ae68d6825b52ec006f385a94d1d","tier":null,"role":"editor"}],"comments":[],"custom_thumbnail":false,"trashed":false,"lapsed":null,"subscribed":null,"forks":3,"collection":{"id":"40c8ef5af1a51aff","slug":"distributions","title":"Statistical distributions","owner":{"id":"7592a7ab121f74a0","login":"mattiasvillani","name":"Mattias Villani"}},"type":"notebook","body":{"title":"F-distribution","cells":[{"id":38,"value":"# F-distribution","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","output":"","database":"$undefined"},{"id":629,"value":"

${tex`X\\sim F(\\nu_1,\\nu_2)`}, for ${tex`X>0`}","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","output":"","database":"$undefined"},{"id":496,"value":"## Probability density function and moments","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","output":"","database":"$undefined"},{"id":118,"value":"viewof params = Inputs.form([\n Inputs.range([1, 20], {value: 4, step: 1, label: tex`\\nu_1`}),\n Inputs.range([1, 20], {value: 4, step: 1, label: tex`\\nu_2`}),\n Inputs.range([0.01, 10], {value: 2, step: 0.01, label: \"quantile:\"})\n ])","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":461,"value":"dist_quantile = tex`\\text{If } X \\sim F(${nu1},${nu2}) \\text{ then } `","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":555,"value":"$33","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":305,"value":"plt_pdf = Plot.plot({\n color: {\n legend: true\n },\n x: {\n label: \"x\",\n axis: true\n },\n y: {\n label: \"f(x)\",\n //axis: false,\n //domain: [0,jstat.chisquare.pdf(mode, nu1, nu2)]\n },\n marks: [\n Plot.ruleY([0]),\n Plot.ruleX([0]),\n Plot.line(fpdf, {x: \"x\", y: \"pdf\", stroke : \"purple\", strokeWidth: 2}),\n Plot.areaY(fpdf, {filter: d => d.x <= quantile, x: \"x\", y: \"pdf\", fill: \"purple\", opacity: 0.2})\n ]\n })","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":610,"value":"## Relations to other distributions","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","output":"","database":"$undefined"},{"id":612,"value":">

If ${tex`X_1 \\sim \\chi^2_{\\nu_1}`} and ${tex`X_2 \\sim \\chi^2_{\\nu_2}`} are independent then\n>

${tex`\\frac{X_1/\\nu_1}{X_2/\\nu_2} \\sim F(\\nu_1,\\nu_2)`}

\n\n ","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","output":"","database":"$undefined"},{"id":109,"value":"
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\n\n---\n## 🙈 Appendix 🙈\n\nHere is some code and data details that you probably do not want to read.","pinned":false,"hidden":false,"mode":"md","author":"$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":654,"value":"nu1 = params[0]","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":656,"value":"nu2 = params[1]","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":662,"value":"quantile = params[2]","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":17,"value":"fpdf = {\n const x = d3.range(0.01, jstat.centralF.inv( 0.99, nu1, nu2 ), 0.01);\n const pdf = x.map(x => ({x: x, pdf: jstat.centralF.pdf(x, nu1, nu2 )}));\n return pdf\n}","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":19,"value":"fcdf = jstat.centralF.cdf(quantile, nu1, nu2);","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":"$W34","stdlib":"1","titling":"auto"}} 32:"$31:body"