2b:null 2c:["$","title",null,{"children":"Zero-inflated Poisson distribution | Mattias Villani | Observable"}] 31:[] 2f:{"id":"b4b1a158502248ad","slug":"zero-inflated-poisson-distribution","title":"Zero-inflated Poisson distribution","thumbnail":"8e0f2e45cc2387e473892e15169285aa35ede933a2d18990a173b0e88a6cea7c","access_level":"public","create_time":"$D2023-06-29T22:00:42.985Z","update_time":"$D2024-02-05T19:28:30.405Z","view_time":null,"first_public_time":"$D2023-06-29T22:00:42.985Z","fork":{"id":"93ddcc9cbf763f75","slug":"poisson-distribution","title":"Poisson 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":868,"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":"Zero-inflated Poisson distribution","cells":[{"id":38,"value":"# Zero-inflated Poisson distribution","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined"},{"id":780,"value":"disttext = md`

${tex`X\\sim \\operatorname{ZIPoisson}(\\pi, \\lambda)`}, for ${tex`X \\in \\{0,1,2,\\ldots \\}`}`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":828,"value":"The probability mass function of a ${tex`X\\sim \\operatorname{ZIPoisson}(\\pi, \\lambda)`} random variable is\n\n${tex`f(x)= \n\n\\begin{cases}\n\\pi + (1-\\pi)e^{-\\lambda} \\,\\,\\,\\,\\,\\, \\text{ for } x = 0\\\\\n(1-\\pi)\\frac{\\lambda^x e^{-\\lambda}}{x!}\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\; \\text{ for } x = 1,2,\\ldots,\n\\end{cases}\n`}\n\nso that the distribution is a mixture of a point mass at zero with probability ${tex`\\pi`} and a ${tex`\\operatorname{Poisson}(\\lambda)`} with probability ${tex`1-\\pi`}.","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined"},{"id":118,"value":"viewof params = Inputs.form([\n Inputs.range([0, 1], {value: 0.5, step: 0.01, label: tex`\\pi:`}),\n Inputs.range([0.1, 10], {value: 2, step: 0.1, label: tex`\\lambda:`}),\n Inputs.range([0, 10], {value: 2, step: 1, label: \"Quantile:\"})\n ])","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":461,"value":"dist_quantile = tex`\\text{If } X \\sim \\operatorname{ZIPoisson}(${pi},${lambda}) \\text{ then } `","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":555,"value":"moments = tex`\n\\begin{aligned}\n&E( X) = \\lambda(1-\\pi) = ${mean.toPrecision(3)} \\\\[1em]\n&Var( X) = \\lambda(1-\\pi)(1 + \\pi\\lambda) = ${variance.toPrecision(3)} \\\\[1em]\n&P( X\\leq ${quantile}) =${zipoiscdf.toPrecision(4)}\n\\end{aligned}\n`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$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(zipoispdf,{x: \"x\", y: \"pdf\", fill: \"#C1E1C1\", strokeWidth: 0, opacity: 1,\n title: (d) => `P(X=${d.x}) = ${(d.pdf).toPrecision(4)}`}),\n Plot.barY(zipoispdf, {filter: d => d.x <= quantile, x: \"x\", y: \"pdf\", fill: \"#50C878\", opacity: 1,\n title: (d) => `P(X=${d.x}) = ${(d.pdf).toPrecision(4)}`})\n ]\n })","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":109,"value":"
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\n\n---\n## 🙈 Appendix 🙈\n\nJust some code and details that you probably do not want to read.","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined"},{"id":221,"value":"#### Imports","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined"},{"id":26,"value":"jstat = require('jstat')","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":764,"value":"import {Plot} from \"@mkfreeman/plot-tooltip\"","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":223,"value":"#### Data","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined"},{"id":797,"value":"pi = params[0]","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":799,"value":"lambda = params[1]","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":801,"value":"quantile = params[2]","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":813,"value":"mean = (1-pi)*lambda","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":815,"value":"variance = lambda*(1-pi)*(1 + pi*lambda)","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":17,"value":"zipoispdf = {\n const x = d3.range(0, lambda + 4*Math.sqrt(lambda), 1);\n const data = x.map(x => ({x: x, pdf: pi*(x==0) + (1-pi)*jstat.poisson.pdf(x, lambda)}));\n return data\n}","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":19,"value":"zipoiscdf = pi + (1-pi)*jstat.poisson.cdf(quantile, lambda);","pinned":false,"hidden":false,"mode":"ojs","author":"$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"