2a:null 2b:["$","title",null,{"children":"Weibull distribution | Mattias Villani | Observable"}] 31:[] 2e:{"id":"882386ab35e15d86","slug":"weibull-distribution","title":"Weibull distribution","thumbnail":"049bcb4b8a0b4a4374153787f289b421128ce258fe775aea119eac3294ea960a","access_level":"public","create_time":"$D2023-04-27T09:45:20.840Z","update_time":"$D2025-02-20T13:39:09.397Z","view_time":null,"first_public_time":"$D2023-04-27T09:45:20.840Z","fork":{"id":"78f34b5c26815cb5","slug":"lognormal-distribution","title":"LogNormal distribution","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":804,"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":2,"collection":{"id":"40c8ef5af1a51aff","slug":"distributions","title":"Statistical distributions","owner":{"id":"7592a7ab121f74a0","login":"mattiasvillani","name":"Mattias Villani"}},"type":"notebook","body":{"title":"Weibull distribution","cells":[{"id":38,"value":"# Weibull distribution","pinned":false,"hidden":false,"mode":"md"},{"id":629,"value":"

${tex`X\\sim \\mathrm{Weibull}(\\lambda,k)`}, for ${tex`X>0`}","pinned":false,"hidden":false,"mode":"md"},{"id":770,"value":"The density function of a ${tex`X \\sim \\mathrm{Weibull}(\\lambda,k)`} random variable is \n\n${tex`f(x)=\\frac{k}{\\lambda}\\left(\\frac{x}{\\lambda}\\right)^{k-1}e^{-(x/\\lambda)^k}`} for ${tex`x>0`}","pinned":false,"hidden":false,"mode":"md"},{"id":118,"value":"viewof params = Inputs.form([\n Inputs.range([0.0, 10], {value: 1, step: 0.1, label: tex`\\lambda`}),\n Inputs.range([0.0, 10], {value: 1, step: 0.1, label: tex`k`}),\n Inputs.range([0, 10], {value: 1, step: 0.1, label: \"quantile:\"})\n ])","pinned":false,"hidden":false,"mode":"ojs"},{"id":461,"value":"dist_quantile = tex`\\text{If } X \\sim \\mathrm{Weibull}(${lambda},${(k).toPrecision(2)}^2) \\text{ then } `","pinned":false,"hidden":false,"mode":"ojs"},{"id":555,"value":"moments = {\n \n return tex`\n \\begin{aligned}\n &E( X) = \\lambda \\Gamma\\Big(1+\\frac{1}{k}\\Big) = ${(mean).toPrecision(3)} \\\\[1em]\n &Var( X) = \\lambda^2 \\left(\\Gamma\\left(1+\\frac{2}{k}\\right) - \\Gamma\\left(1+\\frac{1}{k}\\right)^2\\right) = ${(variance).toPrecision(3)} \\\\[1em]\n &P( X\\leq ${quantile}) =${cdf.toPrecision(4)}\n \\end{aligned}\n `\n}","pinned":false,"hidden":false,"mode":"ojs"},{"id":305,"value":"plt_pdf = Plot.plot({\n color: {\n legend: true\n },\n title: \"Weibull density\",\n x: {\n label: \"x\",\n axis: true\n },\n y: {\n label: \"f(x)\"\n },\n marks: [\n Plot.ruleY([0]),\n Plot.ruleX([0]),\n Plot.line(pdf, {x: \"x\", y: \"pdf\", stroke : \"green\", strokeWidth: 2}),\n Plot.areaY(pdf, {filter: d => d.x <= quantile, x: \"x\", y: \"pdf\", fill: \"green\", opacity: 0.2})\n ]\n })","pinned":false,"hidden":false,"mode":"ojs"},{"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"},{"id":221,"value":"#### Imports","pinned":false,"hidden":false,"mode":"md"},{"id":26,"value":"jstat = require('jstat')","pinned":false,"hidden":false,"mode":"ojs"},{"id":746,"value":"mathjs = require('mathjs')","pinned":false,"hidden":false,"mode":"ojs"},{"id":223,"value":"#### Data","pinned":false,"hidden":false,"mode":"md"},{"id":654,"value":"lambda = params[0]","pinned":false,"hidden":false,"mode":"ojs"},{"id":656,"value":"k = params[1]","pinned":false,"hidden":false,"mode":"ojs"},{"id":662,"value":"quantile = params[2]","pinned":false,"hidden":false,"mode":"ojs"},{"id":706,"value":"mean = lambda*mathjs.gamma(1 + 1/k)","pinned":true,"hidden":false,"mode":"ojs"},{"id":709,"value":"variance = lambda**2*( mathjs.gamma(1 + 2/k) - mathjs.gamma(1+ 1/k)**2 )","pinned":true,"hidden":false,"mode":"ojs"},{"id":17,"value":"pdf = {\n const x = d3.range(0.01, 10, 0.01); // use jstat.lognormal.inv( 0.999, mu, sigma ) as upper if needed\n const pdf = x.map(x => ({x: x, pdf: jstat.weibull.pdf(x, lambda, k )}));\n return pdf\n}","pinned":false,"hidden":false,"mode":"ojs"},{"id":19,"value":"cdf = jstat.weibull.cdf(quantile, lambda, k);","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"