2b:null 2c:["$","title",null,{"children":"Laplace distribution | Mattias Villani | Observable"}] 31:[] 2f:{"id":"3cd309cc51f8e630","slug":"laplace-distribution","title":"Laplace distribution","thumbnail":"4238a464c890328e983b91b3347263d3096eef629fcd8a2be6dad38f2156f40f","access_level":"public","create_time":"$D2023-10-08T07:48:47.381Z","update_time":"$D2024-09-05T11:01:53.617Z","view_time":null,"first_public_time":"$D2023-10-08T07:48:47.381Z","fork":{"id":"4594bdcee9954fb1","slug":"student-t-distribution","title":"Student-t distribution (three-parameter)","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":614,"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":1,"collection":{"id":"40c8ef5af1a51aff","slug":"distributions","title":"Statistical distributions","owner":{"id":"7592a7ab121f74a0","login":"mattiasvillani","name":"Mattias Villani"}},"type":"notebook","body":{"title":"Laplace distribution","cells":[{"id":38,"value":"# Laplace distribution","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined"},{"id":523,"value":"

${tex`X \\sim \\mathrm{Laplace}(\\mu,b)`}, for ${tex`-\\infty \\leq X \\leq \\infty`}","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined"},{"id":406,"value":"The density of a ${tex`X \\sim \\mathrm{Laplace}(\\mu,b)`} random variable is\n\n${tex`f(x) = \\frac{1}{2b}\\exp \\left( - \\frac{ \\vert x - \\mu \\vert}{b} \\right)`}","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined"},{"id":408,"value":"moments = tex`\n\\begin{aligned}\n&\\mathbb{E}( X) = \\mu = ${mean.toFixed(2)} \\\\[1em]\n&\\mathbb{V}( X) = 2b^2 = ${stdev.toFixed(2)}\n\\end{aligned}\n`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":118,"value":"viewof inputs = Inputs.form([\n Inputs.range([-5, 5], {value: 0, step: 0.1, label: tex`\\text{location }\\mu`}),\n Inputs.range([0.1, 5], {value: 1, step: 0.01, label: tex`\\text{scale }b`}),\n Inputs.range([-10, 10], {value: -1.96, step: 0.01, label: \"quantile\"})\n ])","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":450,"value":"viewof shownormal = Inputs.toggle({value: false, label: \"show normal\"})","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":462,"value":"tailprobs = \n{\n var tailprobs;\n if (shownormal){\n tailprobs = tex`\\text{Normal distribution: } P(X\\leq ${quantile})=${normcdfProb.toPrecision(4)} \\\\ \\text{Laplace distribution: } P(X\\leq ${quantile})=${laplacecdfProb.toPrecision(4)}`\n }\n else{\n tailprobs = tex`\\text{Laplace distribution: } P(X\\leq ${quantile})=${laplacecdfProb.toPrecision(4)}`\n }\n return tailprobs;\n}","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":305,"value":"plt = Plot.plot({\n color: {\n legend: shownormal\n },\n x: {\n label: \"x\",\n axis: true\n },\n y: {\n axis: true,\n domain: [0,1.05*d3.max([jstat.normal.pdf(muMatched, muMatched, sigmaMatched), laplacepdf(mu,mu,b)])]\n },\n marks: [\n Plot.ruleY([0]),\n Plot.line(data, {filter: d => d.dist == \"laplace\", x: \"x\", y: \"pdf\", stroke : \"dist\", strokeWidth: 2}),\n Plot.line(data, {filter: d => d.dist == \"normal (moments matched)\" && shownormal, x: \"x\", y: \"pdf\", stroke : \"dist\", strokeWidth: 2}),\n //Plot.areaY(data, {filter: d => d.x <= quantile && d.dist == \"normal\" && shownormal, x: \"x\", y: \"pdf\", fill: \"steelblue\", opacity: 0.2}),\n Plot.areaY(data, {filter: d => d.x <= quantile && d.dist == \"laplace\", x: \"x\", y: \"pdf\", fill: \"dist\", opacity: 0.2}),\n \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 only code and details that you probably don't 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":546,"value":"mu = inputs[0]","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":553,"value":"b = inputs[1]","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":557,"value":"quantile = inputs[2]","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":565,"value":"mean = mu","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":567,"value":"stdev = Math.sqrt(2*b**2)","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":17,"value":"data = {\n const x = d3.range(-10, 10, 0.01);\n const normpdf = x.map(x => ({x: x, pdf: jstat.normal.pdf(x, muMatched, sigmaMatched), dist: \"normal (moments matched)\"}));\n const laplacepdfval = x.map(x => ({x: x, pdf: laplacepdf(x, mu, b), dist: \"laplace\"}));\n const data = normpdf.concat(laplacepdfval)\n return data\n}","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":595,"value":"muMatched = mu","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":596,"value":"sigmaMatched = Math.sqrt(2)*b","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":19,"value":"normcdfProb = jstat.normal.cdf(quantile, muMatched, sigmaMatched);","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":582,"value":"function laplacepdf(x,mu,b){\n return (1/(2*b))*Math.exp(- Math.abs((x-mu)/b) )\n}","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":23,"value":"function laplacecdf(x, mu, b){\n if (x <= mu){\n var cdf = 0.5*Math.exp((x-mu)/b)\n }else{\n var cdf = 1 - 0.5*Math.exp(-(x-mu)/b)\n }\n return cdf\n}","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":575,"value":"laplacecdfProb = laplacecdf(quantile, mu, 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"