2b:null 2c:["$","title",null,{"children":"Pareto distribution | Mattias Villani | Observable"}] 31:T40c,moments = { if (alpha<=1){ return tex` \begin{aligned} &E( X) =\infty \text{ if } \alpha\leq 1 \\[1em] &Var( X) = \text{ does not exist if } \alpha \leq 1 \\[1em] &P( X\leq ${quantile}) =${paretocdf.toPrecision(4)} \end{aligned} ` } else{ if (alpha <=2){ return tex` \begin{aligned} &E( X) =\frac{\alpha x_m}{\alpha -1} = ${(alpha*xm/(alpha-1)).toPrecision(3)} \text{ if } \alpha > 1\\[1em] &Var( X) = \infty \text{ if } \alpha \leq 2 \\[1em] &P( X\leq ${quantile}) =${paretocdf.toPrecision(4)} \end{aligned} ` } else{ return tex` \begin{aligned} &E( X) = \frac{\alpha x_m}{\alpha -1} = ${(alpha*xm/(alpha-1)).toPrecision(3)} \text{ if } \alpha > 1\\[1em] &Var( X) = \frac{\alpha x_m^2}{(\alpha-1)^2(\alpha-2)} = ${(alpha**2*xm/((alpha-1)**2*(alpha-2))).toPrecision(3)} \text{ if } \alpha > 2 \\[1em] &P( X\leq ${quantile}) =${paretocdf.toPrecision(4)} \end{aligned} ` } } }32:[] 2f:{"id":"a962631c67dd3941","slug":"pareto-distribution","title":"Pareto distribution","thumbnail":"e3619487aa2a1713c74229fbad94cdc0e78a4182c328870c2f0dc2a3087e41da","access_level":"public","create_time":"$D2023-04-25T08:19:06.577Z","update_time":"$D2023-09-04T09:19:27.010Z","view_time":null,"first_public_time":"$D2023-04-25T08:19:06.577Z","fork":{"id":"372affbdf9c3bc62","slug":"f-distribution","title":"F-distribution","owner":{"id":"7592a7ab121f74a0","login":"mattiasvillani","name":"Mattias Villani"}},"stars":0,"starred":false,"owner":{"type":"team","id":"7592a7ab121f74a0","login":"mattiasvillani","name":"Mattias Villani","avatar_url":"https://avatars-next.observableusercontent.com/a440aaf39a3abefb86cbcc758fd7a5ab42555ae68d6825b52ec006f385a94d1d","tier":"pro_2024"},"version":743,"publish_version":null,"role":null,"authored":false,"authors":[{"type":"individual","id":"56499be8cf43f0ef","login":"mattiasvillani","name":"Mattias Villani","avatar_url":"https://avatars-next.observableusercontent.com/a440aaf39a3abefb86cbcc758fd7a5ab42555ae68d6825b52ec006f385a94d1d","tier":null}],"members":[{"type":"individual","id":"56499be8cf43f0ef","login":"mattiasvillani","name":"Mattias Villani","avatar_url":"https://avatars-next.observableusercontent.com/a440aaf39a3abefb86cbcc758fd7a5ab42555ae68d6825b52ec006f385a94d1d","tier":null,"role":"editor"}],"comments":[],"custom_thumbnail":false,"trashed":false,"subscribed":null,"forks":0,"collection":{"id":"40c8ef5af1a51aff","slug":"distributions","title":"Statistical distributions","owner":{"id":"7592a7ab121f74a0","login":"mattiasvillani","name":"Mattias Villani"}},"type":"notebook","body":{"title":"Pareto distribution","cells":[{"id":38,"value":"# Pareto distribution","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined"},{"id":629,"value":"

${tex`X\\sim \\mathrm{Pareto}(x_m,\\alpha)`}, for ${tex`X\\geq x_m`}","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined"},{"id":496,"value":"## Probability density function and moments","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined"},{"id":118,"value":"viewof params = Inputs.form([\n Inputs.range([0.1, 5], {value: 1, step: 0.1, label: tex`x_{m}`}),\n Inputs.range([1, 10], {value: 4, step: 0.1, label: tex`\\alpha`})\n ])","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":699,"value":"viewof quantile = Inputs.range([xm, 10], {value: xm + 1, step: 0.01, label: \"quantile:\"})","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":461,"value":"dist_quantile = tex`\\text{If } X \\sim \\mathrm{Pareto}(${xm}, ${alpha}) \\text{ then } `","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":555,"value":"$31","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":305,"value":"plt_pdf = Plot.plot({\n color: {\n legend: true\n },\n x: {\n label: \"x\",\n axis: true\n },\n y: {\n label: \"f(x)\",\n //axis: false,\n //domain: [0,jstat.chisquare.pdf(mode, nu1, nu2)]\n },\n marks: [\n Plot.ruleY([0]),\n Plot.ruleX([0]),\n Plot.line(paretopdf, {x: \"x\", y: \"pdf\", stroke : \"orange\", strokeWidth: 2}),\n Plot.areaY(paretopdf, {filter: d => d.x <= quantile, x: \"x\", y: \"pdf\", fill: \"orange\", opacity: 0.2})\n ]\n })","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":109,"value":"
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\n\n---\n## 🙈 Appendix 🙈\n\nHere is some code and data details that you probably do not want to read.","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined"},{"id":221,"value":"#### Imports","pinned":false,"hidden":false,"mode":"md","author":"$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":"$undefined"},{"id":656,"value":"xm = params[0]","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":654,"value":"alpha = params[1]","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":17,"value":"paretopdf = {\n const x = d3.range(xm, 10, 0.01); //jstat.pareto.inv( 0.99, xm, alpha )\n const pdf = x.map(x => ({x: x, pdf: jstat.pareto.pdf(x, xm, alpha )}));\n return pdf\n}","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":19,"value":"paretocdf = jstat.pareto.cdf(quantile, xm, alpha);","pinned":false,"hidden":false,"mode":"ojs","author":"$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":"$undefined"}],"files":"$W32","stdlib":"1","titling":"auto"}} 30:"$2f:body"