2a:null 2b:["$","title",null,{"children":"Beta distribution | Mattias Villani | Observable"}] 31:[] 2e:{"id":"0ce89a02b7e0a2f4","slug":"beta-distribution","title":"Beta distribution","thumbnail":"e1885be4b9873b8a7d1af61c7b96bc6d0b1bd1f1899771bbff1c098d00626b21","access_level":"public","create_time":"$D2023-01-31T21:52:03.361Z","update_time":"$D2026-01-16T22:55:53.281Z","view_time":null,"first_public_time":"$D2023-01-31T21:52:03.361Z","fork":{"id":"04eba2cc11169bfb","slug":"beta-fordelningen","title":"Beta-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":794,"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":5,"collection":{"id":"40c8ef5af1a51aff","slug":"distributions","title":"Statistical distributions","owner":{"id":"7592a7ab121f74a0","login":"mattiasvillani","name":"Mattias Villani"}},"type":"notebook","body":{"title":"Beta distribution","cells":[{"id":38,"value":"# Beta distribution","pinned":false,"hidden":false,"mode":"md"},{"id":662,"value":"disttext = {\n var text;\n if (parametrization == 'traditional'){\n text = md`

${tex`X\\sim \\operatorname{Beta}(\\alpha,\\beta)`}, for ${tex`X \\in (0,1)`}`\n }else{\n text = md`

${tex`X\\sim \\operatorname{Beta}(\\mu,\\phi)`}, for ${tex`X \\in (0,1)`}`}\nreturn text;\n}","pinned":false,"hidden":false,"mode":"ojs"},{"id":786,"value":"The Beta distribution has two commonly used parameterizations: \n- the traditional one with parameters ${tex`\\alpha>0`} and ${tex`\\beta>0`}\n- a parameterization where the mean ${tex`0<\\mu<1`} is an explicit parameter.","pinned":false,"hidden":false,"mode":"md"},{"id":724,"value":"viewof parametrization = Inputs.radio([\"traditional\", \"mean\"], {label: \"Parameterization:\", value: \"traditional\"})","pinned":false,"hidden":false,"mode":"ojs"},{"id":496,"value":"pdftext = {\n var pdftext;\n if (parametrization == 'traditional'){\n pdftext = md`${tex`f(x)=\\frac{1}{\\Beta(\\alpha,\\beta)} x^{\\alpha-1}(1-x)^{\\beta-1}`}`\n }else{\n pdftext = md`${tex`f(x)=\\frac{1}{\\Beta(\\mu\\phi,\\,(1-\\mu)\\phi)} x^{\\mu\\phi-1}(1-x)^{(1-\\mu)\\phi-1}`}`\n }\nreturn pdftext;\n}\n","pinned":false,"hidden":false,"mode":"ojs"},{"id":118,"value":"viewof params = {\n\nvar form;\n if (parametrization == 'traditional'){\n form = Inputs.form([\n Inputs.range([0.1, 20], {value: 3, step: 0.1, label: tex`\\alpha`}),\n Inputs.range([0.1, 20], {value: 3, step: 0.1, label: tex`\\beta`}),\n Inputs.range([0, 1], {value: 0.2, step: 0.01, label: \"Quantile:\"})\n ])\n }\n else if (parametrization == 'mean'){\n form = Inputs.form([\n Inputs.range([0.001, 0.999], {value: 0.5, step: 0.001, label: tex`\\mu`}),\n Inputs.range([0.01, 1000], {value: 6, step: 0.1, label: tex`\\phi`}),\n Inputs.range([0, 1], {value: 0.2, step: 0.01, label: \"Quantile:\"})\n ])\n }\n \n return form;\n}","pinned":false,"hidden":false,"mode":"ojs"},{"id":461,"value":"dist_quantile = tex`\\text{If } X \\sim \\operatorname{Beta}(${params[0]},${params[1]}) \\text{ then } `","pinned":false,"hidden":false,"mode":"ojs"},{"id":555,"value":"moments = {\n \n if (parametrization == 'traditional'){\n return tex`\n \\begin{aligned}\n &\\mathbb{E}(X) =\\frac{\\alpha}{\\alpha + \\beta} = ${(params[0]/(params[0] + params[1])).toPrecision(2)} \\\\[1em]\n &\\mathbb{V}(X) = \\frac{\\alpha \\beta}{(\\alpha + \\beta)^2(\\alpha + \\beta + 1)} = ${((params[0]*params[1])/((params[0] + params[1])**2*(params[0] + params[1] +1 ))).toPrecision(2)} \\\\[1em]\n &\\mathrm{Pr}( X\\leq ${params[2]}) =${betacdf.toPrecision(4)}\n \\end{aligned}\n `\n }else{\n return tex`\n \\begin{aligned}\n &\\mathbb{E}(X) =\\mu = ${params[0].toPrecision(2)} \\\\[1em]\n &\\mathbb{V}(X) = \\frac{\\mu(1-\\mu)}{1 + \\phi} = ${((params[0]*(1-params[0]))/(1+params[1])).toPrecision(2)} \\\\[1em]\n &\\mathrm{Pr}( X\\leq ${params[2]}) =${betacdf.toPrecision(4)}\n \\end{aligned}\n `\n }\n}","pinned":false,"hidden":false,"mode":"ojs"},{"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.beta.pdf((alpha-1)/(alpha+beta-2), alpha, beta)]\n },\n marks: [\n Plot.ruleY([0]),\n Plot.ruleX([0]),\n Plot.line(betapdf, {x: \"x\", y: \"pdf\", stroke : col, strokeWidth: 2}),\n Plot.areaY(betapdf, {filter: d => d.x <= params[2], x: \"x\", y: \"pdf\", fill: col, opacity: 0.2})\n ]\n })","pinned":false,"hidden":false,"mode":"ojs"},{"id":109,"value":"
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\n\n---\n## 🙈 Appendix 🙈\n\nJust some code and details that you probably want to ignore.","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":"betapdf = {\n const x = d3.range(0, 1.01, 0.001);\n const pdf = x.map(x => ({x: x, pdf: jstat.beta.pdf(x,alpha, beta)}));\n return pdf\n}","pinned":false,"hidden":false,"mode":"ojs"},{"id":19,"value":"betacdf = jstat.beta.cdf(params[2], alpha, beta);","pinned":false,"hidden":false,"mode":"ojs"},{"id":745,"value":"distparam = {\n let alpha, beta;\n if (parametrization == 'traditional'){\n alpha = params[0];\n beta = params[1];\n }else{\n alpha = params[0]*params[1];\n beta = (1-params[0])*params[1]\n }\n return {alpha: alpha, beta: beta}\n}","pinned":false,"hidden":false,"mode":"ojs"},{"id":751,"value":"alpha = distparam.alpha","pinned":false,"hidden":false,"mode":"ojs"},{"id":753,"value":"beta = distparam.beta","pinned":false,"hidden":false,"mode":"ojs"},{"id":766,"value":"col = {\n let col\n if (parametrization == 'traditional'){\n col = \"cornflowerblue\"\n }else{\n col = \"orange\"\n }\n return col\n}","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"