2a:null 2b:["$","title",null,{"children":"LogNormal distribution | Mattias Villani | Observable"}] 31:[] 2e:{"id":"78f34b5c26815cb5","slug":"lognormal-distribution","title":"LogNormal distribution","thumbnail":"2b01e18583d4100b09ff1274e59f4a0aac73ed87e113435488991376955d4b9b","access_level":"public","create_time":"$D2023-04-25T15:27:56.576Z","update_time":"$D2025-09-24T12:50:00.649Z","view_time":null,"first_public_time":"$D2023-04-25T15:27:56.576Z","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":739,"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":"LogNormal distribution","cells":[{"id":38,"value":"# LogNormal distribution","pinned":false,"hidden":false,"mode":"md"},{"id":629,"value":"

${tex`X\\sim \\mathrm{logN}(\\mu,\\sigma^2)`}, for ${tex`X>0`}","pinned":false,"hidden":false,"mode":"md"},{"id":496,"value":"The density function of a ${tex`X\\sim \\mathrm{logN}(\\mu,\\sigma^2)`} random variable is\n\n${tex`f(x)=\\frac 1 {x\\sigma\\sqrt{2\\pi}}\\ \\exp\\left( - \\frac{\\left(\\ln x -\\mu\\right)^2}{2\\sigma^2}\\right)`}","pinned":false,"hidden":false,"mode":"md"},{"id":118,"value":"viewof params = Inputs.form([\n Inputs.range([-1, 1], {value: 0, step: 0.1, label: tex`\\mu`}),\n Inputs.range([0.1, 1], {value: 0.5, step: 0.01, label: tex`\\sigma`}),\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{LogN}(${mu},${(sigma).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) = \\exp(\\mu + \\frac{\\sigma^2}{2}) = ${(mean).toPrecision(3)} \\\\[1em]\n &Var( X) = (\\exp(\\sigma^2)-1)\\exp(2\\mu +\\sigma^2) = ${(variance).toPrecision(3)} \\\\[1em]\n &P( X\\leq ${quantile}) =${lognormalcdf.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 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(lognormalpdf, {x: \"x\", y: \"pdf\", stroke : \"purple\", strokeWidth: 2}),\n Plot.areaY(lognormalpdf, {filter: d => d.x <= quantile, x: \"x\", y: \"pdf\", fill: \"purple\", opacity: 0.2})\n ]\n })","pinned":false,"hidden":false,"mode":"ojs"},{"id":109,"value":"
\n
\n
\n
\n
\n
\n
\n
\n
\n
\n
\n
\n
\n
\n
\n
\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":223,"value":"#### Data","pinned":false,"hidden":false,"mode":"md"},{"id":654,"value":"mu = params[0]","pinned":false,"hidden":false,"mode":"ojs"},{"id":656,"value":"sigma = params[1]","pinned":false,"hidden":false,"mode":"ojs"},{"id":662,"value":"quantile = params[2]","pinned":false,"hidden":false,"mode":"ojs"},{"id":706,"value":"mean = Math.exp(mu + 0.5*sigma**2)","pinned":true,"hidden":false,"mode":"ojs"},{"id":709,"value":"variance = (Math.exp(sigma**2)-1)*Math.exp(2*mu+sigma**2)","pinned":true,"hidden":false,"mode":"ojs"},{"id":17,"value":"lognormalpdf = {\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.lognormal.pdf(x, mu, sigma )}));\n return pdf\n}","pinned":false,"hidden":false,"mode":"ojs"},{"id":19,"value":"lognormalcdf = jstat.lognormal.cdf(quantile, mu, sigma);","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"