29:null 2a:["$","title",null,{"children":"Poisson distribution | Mattias Villani | Observable"}] 2f:[] 2d:{"id":"93ddcc9cbf763f75","slug":"poisson-distribution","title":"Poisson distribution","thumbnail":"2c768af3806cea227c96b6696e0aedb0c17e02a1d3775b5dd76ffe52f2f2c82a","access_level":"public","create_time":"$D2023-01-31T21:54:25.793Z","update_time":"$D2024-02-16T16:41:47.457Z","view_time":null,"first_public_time":"$D2023-01-31T21:54:25.793Z","has_unpublished_edits":null,"fork":{"id":"ead37e06dd3cddac","slug":"poissonfordelning","title":"Poissonfördelning","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":"education_2026"},"version":808,"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":5,"collection":{"id":"40c8ef5af1a51aff","slug":"distributions","title":"Statistical distributions","owner":{"id":"7592a7ab121f74a0","login":"mattiasvillani","name":"Mattias Villani"}},"type":"notebook","body":{"title":"Poisson distribution","cells":[{"id":38,"value":"# Poisson distribution","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","output":"","database":"$undefined"},{"id":780,"value":"disttext = md`

${tex`X\\sim \\operatorname{Poisson}(\\lambda)`}, for ${tex`X \\in \\{0,1,2,\\ldots \\}`}`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":496,"value":"A random variable following the Poisson distribution has probability mass function\n\n${tex.block`P(X=x)= \\frac{\\lambda^x e^{-\\lambda}}{x!}`}","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","output":"","database":"$undefined"},{"id":118,"value":"viewof params = Inputs.form([\n Inputs.range([0.1, 20], {value: 2, step: 0.1, label: tex`\\lambda:`}),\n Inputs.range([1, 40], {value: 1, step: 1, label: \"quantile:\"})\n ])","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":461,"value":"dist_quantile = tex`\\text{If } X \\sim \\operatorname{Poisson}(${params[0]}) \\text{ then } `","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":555,"value":"moments = tex`\n\\begin{aligned}\n&\\mathrm{E}( X) = \\lambda = ${params[0].toPrecision(3)} \\\\[0.5em]\n&\\mathrm{Var}( X) = \\lambda = ${params[0].toPrecision(3)} \\\\[0.5em]\n&\\mathrm{P}( X\\leq ${params[1]}) =${poiscdf.toPrecision(4)}\n\\end{aligned}\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: false\n },\n x: {\n label: \"x\",\n axis: true\n },\n y: {\n label: \"f(x)\",\n axis: true\n },\n tooltip: {\n fill: \"#097969\",\n stroke: \"#097969\",\n opacity: 1,\n },\n marks: [\n Plot.ruleY([0]),\n Plot.barY(poispdf,{x: \"x\", y: \"pdf\", fill: \"#C1E1C1\", strokeWidth: 0, opacity: 1,\n title: (d) => `P(X=${d.x}) = ${(d.pdf).toPrecision(4)}`}),\n Plot.barY(poispdf, {filter: d => d.x <= params[1], x: \"x\", y: \"pdf\", fill: \"#50C878\", opacity: 1,\n title: (d) => `P(X=${d.x}) = ${(d.pdf).toPrecision(4)}`})\n ]\n })","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":109,"value":"
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\n\n---\n## 🙈 Appendix 🙈\n\nJust some code and details that you probably do not want to read.","pinned":false,"hidden":false,"mode":"md","author":"$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":"$undefined"},{"id":764,"value":"import {Plot} from \"@mkfreeman/plot-tooltip\"","pinned":false,"hidden":false,"mode":"ojs","author":"$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":"poispdf = {\n const x = d3.range(0, params[0] + 4*Math.sqrt(params[0]), 1);\n const data = x.map(x => ({x: x, pdf: jstat.poisson.pdf(x, params[0])}));\n return data\n}","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":19,"value":"poiscdf = jstat.poisson.cdf(params[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":"$W2f","stdlib":"1","titling":"auto"}} 2e:"$2d:body"