24:null 25:["$","title",null,{"children":"Exponential distribution | Mattias Villani | Observable"}] 2a:[] 28:{"id":"1fd0cc7fe678b852","slug":"exponential-distribution","title":"Exponential distribution","thumbnail":"ff2bd4639185eb07f306545111e58c53145f4a06776de4e1d01feb9be8ae8f72","access_level":"public","create_time":"$D2023-05-13T18:51:37.456Z","update_time":"$D2025-05-05T11:51:16.596Z","view_time":null,"first_public_time":"$D2023-05-13T18:51:37.456Z","fork":{"id":"25d835ea95563034","slug":"exponentialfordelningen","title":"Exponentialfördelningen","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":660,"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,"lapsed":null,"subscribed":null,"forks":1,"collection":{"id":"40c8ef5af1a51aff","slug":"distributions","title":"Statistical distributions","owner":{"id":"7592a7ab121f74a0","login":"mattiasvillani","name":"Mattias Villani"}},"type":"notebook","body":{"title":"Exponential distribution","cells":[{"id":38,"value":"# Exponential distribution","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","output":"","database":"$undefined"},{"id":586,"value":"disttext = {\n var text;\n if (parametrization == 'rate'){\n text = md`

${tex`X\\sim \\operatorname{Expon}(\\lambda)`}, for ${tex`X \\in (0,\\infty)`}`\n }else{\n text = md`

${tex`X\\sim \\operatorname{Expon}(\\beta)`}, for ${tex`X \\in (0,\\infty)`}`}\nreturn text;\n}","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":626,"value":"viewof parametrization = Inputs.radio([\"scale\", \"rate\"], {label: \"Parameterization:\", value: \"rate\"})","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":604,"value":"pdftext = {\n var pdftext;\n if (parametrization == 'rate'){\n pdftext = md`${tex`f(x)=\\lambda e^{-\\lambda x }`}`\n }else{\n pdftext = md`${tex`f(x)=\\frac{1}{\\beta} e^{-\\frac{x}{\\beta}}`}`\n }\nreturn pdftext;\n}","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":637,"value":"viewof params = {\n\nvar form;\n if (parametrization == 'scale'){\n form = Inputs.form([\n Inputs.range([0, 10], {value: 1, step: 0.1, label: tex`\\beta:`}),\n Inputs.range([0, 5], {value: 1, step: 0.01, label: \"quantile:\"})\n ])\n }\n else if (parametrization == 'rate'){\n form = Inputs.form([\n Inputs.range([0, 10], {value: 1, step: 0.1, label: tex`\\lambda:`}),\n Inputs.range([0, 10], {value: 1, step: 0.01, label: \"quantile:\"})\n ])\n }\n \n return form;\n}","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":461,"value":"dist_quantile = tex`\\text{If } X \\sim \\operatorname{Expon}(${params[0]}) \\text{ then } `","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":643,"value":"textmoments = {\n var text;\n if (parametrization == 'rate'){\n text = tex`\n\\begin{aligned}\n&E( X) =\\frac{1}{\\lambda} = ${(1/params[0]).toPrecision(3)} \\\\[1em]\n&Var( X) =\\frac{1}{\\lambda^2} = ${(1/params[0]**2).toPrecision(3)} \\\\[1em]\n&P( X\\leq ${params[1]}) =${exponcdf.toPrecision(4)}\n\\end{aligned}\n`}\nelse{\n text = tex`\n\\begin{aligned}\n&E( X) =\\beta = ${(params[0]).toPrecision(3)} \\\\[1em]\n&Var( X) =\\beta^2 = ${(params[0]**2).toPrecision(3)} \\\\[1em]\n&P( X\\leq ${params[1]}) =${exponcdf.toPrecision(4)}\n\\end{aligned}\n`\n}\n return text\n}","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":305,"value":"plt_pdf = Plot.plot({\n color: {\n legend: true\n },\n x: {\n label: \"x\",\n axis: true,\n domain: [0,5]\n },\n y: {\n label: \"f(x)\",\n //axis: false,\n domain: (parametrization == \"rate\") ? [0,jstat.exponential.pdf(0.01, params[0])] : [0,jstat.exponential.pdf(0.01, 1/params[0])]\n },\n marks: [\n Plot.ruleY([0]),\n Plot.ruleX([0]),\n Plot.line(exponpdf, {x: \"x\", y: \"pdf\", \n stroke : (parametrization == \"rate\") ? \"orange\" : \"steelblue\",\n strokeWidth: 2}),\n Plot.areaY(exponpdf, {filter: d => d.x <= params[1], x: \"x\", y: \"pdf\", \n fill: (parametrization == \"rate\") ? \"orange\" : \"steelblue\", \n opacity: 0.2})\n ]\n })","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":109,"value":"
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\n\n---\n## 🙈 Appendix 🙈\n\nHere is just some code and data that you probably want to ignore.","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","output":"","database":"$undefined"},{"id":221,"value":"#### Imports","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","output":"","database":"$undefined"},{"id":26,"value":"jstat = require('jstat')","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":223,"value":"#### Data","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","output":"","database":"$undefined"},{"id":17,"value":"exponpdf = {\n const x = d3.range(Number.EPSILON, 5, 0.001);\n var exponpdf2;\n if (parametrization == \"rate\"){\n exponpdf2 = x.map(x => ({x: x, pdf: jstat.exponential.pdf(x, params[0])}));\n }\n else{\n exponpdf2 = x.map(x => ({x: x, pdf: jstat.exponential.pdf(x, 1/params[0])}));\n }\n return exponpdf2\n}","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":19,"value":"exponcdf = (parametrization == \"rate\")? jstat.exponential.cdf(params[1], params[0]) : jstat.exponential.cdf(params[1], 1/params[0]) ","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":229,"value":"#### Style","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","output":"","database":"$undefined"},{"id":202,"value":"","pinned":false,"hidden":false,"mode":"html","author":"$undefined","summary":"$undefined","output":"","database":"$undefined"},{"id":58,"value":"viewof md_colour = Inputs.color({label: \"Highlight color cells\", value: \"#F5F5DC\"})","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"}],"files":"$W2a","stdlib":"1","titling":"auto"}} 29:"$28:body"