2b:null 2c:["$","title",null,{"children":"Truncated Normal Distribution | Mattias Villani | Observable"}] 31:[] 2f:{"id":"1af26dd5fe3b0845","slug":"truncated-normal-distribution","title":"Truncated Normal Distribution","thumbnail":"828a3bcfc441ca8b35793cf84fae124617dbba87d40202a387238c14e1866f25","access_level":"public","create_time":"$D2023-11-11T11:14:23.202Z","update_time":"$D2023-11-11T11:45:36.686Z","view_time":null,"first_public_time":"$D2023-11-11T11:40:31.016Z","fork":true,"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":767,"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":"Truncated Normal Distribution","cells":[{"id":38,"value":"# Truncated Normal Distribution","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined"},{"id":629,"value":"

${tex`X\\sim N(\\mu,\\sigma^2, a, b)`}, for ${tex`a \\leq X \\leq b`}","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined"},{"id":496,"value":"## Probability density function and moments","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined"},{"id":118,"value":"viewof params = Inputs.form([\n Inputs.range([-5, 5], {value: 0, step: 0.1, label: tex`\\mu`}),\n Inputs.range([Number.EPSILON, 20], {value: 1, step: 0.1, label: tex`\\sigma`}),\n Inputs.range([-5, 5], {value: -2, step: 0.1, label: tex`a`}),\n Inputs.range([-5, 5], {value: 3, step: 0.1, label: tex`b`}),\n Inputs.range([-5, 5], {value: -1, step: 0.01, label: \"quantile:\"})\n ])","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":461,"value":"dist_quantile = tex`\\text{If } X \\sim \\mathrm{TruncNormal}(${mu},${sigma},${a},${b}) \\text{ then } `","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":555,"value":"moments = {\n return tex`\n \\begin{aligned}\n &E( X) = \\mu + \\frac{\\phi(a \\vert \\mu,\\sigma^2) - \\phi(b \\vert \\mu,\\sigma^2)}{\\Phi(b \\vert \\mu,\\sigma^2) - \\Phi(a \\vert \\mu,\\sigma^2)} = ${(mean.toFixed(3))}\\\\[1em]\n &StDev( X) = ${Math.sqrt(variance).toFixed(3)} \\\\[1em]\n &P( X\\leq ${quantile}) =${cdf.toFixed(3)}\n \\end{aligned}\n `\n}","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.line(pdfdata, {x: \"x\", y: \"pdf\", stroke : \"purple\", strokeWidth: 2}),\n Plot.areaY(pdfdata, {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"},{"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":654,"value":"mu = params[0]","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":656,"value":"sigma = params[1]","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":709,"value":"a = params[2]","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":711,"value":"b = params[3]","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":732,"value":"mean = mu + (truncnormalpdf(a, mu, sigma, a, b) - truncnormalpdf(b, mu, sigma, a, b))/normfactor","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":747,"value":"variance = sigma**2*( 1 - (b*truncnormalpdf(b, mu, sigma, a, b) - a*truncnormalpdf(a, mu, sigma, a, b))/normfactor \n - ((truncnormalpdf(a, mu, sigma, a, b) - truncnormalpdf(b, mu, sigma, a, b))/normfactor)**2\n ) ","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":662,"value":"quantile = params[4]","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":740,"value":"normfactor = jstat.normal.cdf(b, mu, sigma) - jstat.normal.cdf(a, mu, sigma)","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":716,"value":"function truncnormalpdf(x, mu, sigma, a, b){\n if (xb) return 0\n return jstat.normal.pdf(x, mu, sigma)/(jstat.normal.cdf(b, mu, sigma) - jstat.normal.cdf(a, mu, sigma))\n}","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":718,"value":"function truncnormalcdf(x, mu, sigma, a, b){\n return (jstat.normal.cdf(x, mu, sigma) - jstat.normal.cdf(a, mu, sigma))/(jstat.normal.cdf(b, mu, sigma) - jstat.normal.cdf(a, mu, sigma))\n}","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":17,"value":"pdfdata = {\n const x = d3.range(-5, 5, 0.01);\n const pdf = x.map(x => ({x: x, pdf: truncnormalpdf(x,mu,sigma,a,b)}));\n return pdf\n}","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":19,"value":"cdf = truncnormalcdf(quantile, mu, sigma, a, b);","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":"$W31","stdlib":"1","titling":"auto"}} 30:"$2f:body"