2a:null 2b:["$","title",null,{"children":"Negative binomial distribution | Mattias Villani | Observable"}] 31:[] 2e:{"id":"d39a8714e5c419e7","slug":"negative-binomial-distribution","title":"Negative binomial distribution","thumbnail":"60644e7ad66cf5a697624fdef3c0c390930d2dc0011afa1274f613c763c5ef8e","access_level":"public","create_time":"$D2023-04-25T18:03:40.782Z","update_time":"$D2025-05-22T09:39:36.931Z","view_time":null,"first_public_time":"$D2023-04-25T18:03:40.782Z","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":897,"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":2,"collection":{"id":"40c8ef5af1a51aff","slug":"distributions","title":"Statistical distributions","owner":{"id":"7592a7ab121f74a0","login":"mattiasvillani","name":"Mattias Villani"}},"type":"notebook","body":{"title":"Negative binomial distribution","cells":[{"id":38,"value":"# Negative binomial distribution","pinned":false,"hidden":false,"mode":"md"},{"id":864,"value":"textdist = {\n var text;\n if (parametrization == 'standard'){\n text = md`

${tex`X\\sim \\operatorname{NegBin}(r,p)`}, for ${tex`X \\in \\{0,1,2,\\ldots \\}`}`\n }\nelse{\n text = md`

${tex`X\\sim \\operatorname{NegBin}(r,\\mu)`}, for ${tex`X \\in \\{0,1,2,\\ldots \\}`}`\n}\n return text\n}","pinned":false,"hidden":false,"mode":"ojs"},{"id":896,"value":"Note: ${tex`X`} counts the number of **failures** until the ${tex`r`}th success.","pinned":false,"hidden":false,"mode":"md"},{"id":819,"value":"viewof parametrization = Inputs.radio([\"standard\", \"mean\"], {label: \"Parameterization:\", value: \"standard\"})","pinned":false,"hidden":false,"mode":"ojs"},{"id":496,"value":"viewof params = {\n\nvar form;\n if (parametrization == 'standard'){\n form = Inputs.form([\n Inputs.range([1, 10], {value: 1, step: 0.1, label: tex`r:`}),\n Inputs.range([0.01, 0.99], {value: 0.5, step: 0.01, label: tex`p:`}),\n Inputs.range([0, 10], {value: 1, step: 1, label: \"Quantile:\"})\n ])\n }\n else if (parametrization == 'mean'){\n form = Inputs.form([\n Inputs.range([1, 1000], {value: 1, step: 0.5, label: tex`r:`}),\n Inputs.range([1, 10], {value: 0.5, step: 0.01, label: tex`\\mu`}),\n Inputs.range([0, 10], {value: 1, step: 1, label: \"Quantile:\"})\n ])\n }\n \n return form;\n}","pinned":false,"hidden":false,"mode":"ojs"},{"id":461,"value":"textmoments = {\n var text;\n if (parametrization == 'standard'){\n text = tex`\n\\text{If } X \\sim \\operatorname{NegBin}(r=${r},p=${params[1].toPrecision(3)}) \\text{ then } \\\\[1em]\n\\begin{aligned}\n&E( X) = \\frac{r(1-p)}{p} = ${mean.toPrecision(3)} \\\\[1em]\n&Var( X) = \\frac{r(1-p)}{p^2} = ${variance.toPrecision(3)} \\\\[1em]\n&P( X\\leq ${params[2]}) =${negbincdf.toPrecision(4)}\n\\end{aligned}\n`}\nelse{\n text = tex`\n\\text{If } X \\sim \\operatorname{NegBin}(r=${r},\\mu=${mean.toPrecision(3)}) \\text{ then } \\\\[1em]\n\\begin{aligned}\n&E( X) = \\mu = ${mean.toPrecision(3)} \\\\[1em]\n&Var( X) = \\mu\\Big(1+\\frac{\\mu}{r}\\Big) = ${variance.toPrecision(3)} \\\\[1em]\n&P( X\\leq ${params[2]}) =${negbincdf.toPrecision(4)}\n\\end{aligned}\n`\n}\n return text\n}\n\n","pinned":false,"hidden":false,"mode":"ojs"},{"id":883,"value":"viewof plotpois = Inputs.toggle({label: \"Plot Poisson(μ)\", value: false})","pinned":false,"hidden":false,"mode":"ojs"},{"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: \"#673147\",\n stroke: \"#673147\",\n opacity: 1,\n },\n marks: [\n Plot.ruleY([0]),\n Plot.barY(negbinpdf,{x: \"x\", y: \"pdf\", fill: \"#AA98A9\", strokeWidth: 0, opacity: .8,\n title: (d) => `P(X=${d.x}) = ${(d.pdf).toPrecision(4)}`}),\n Plot.barY(negbinpdf, {filter: d => d.x <= params[2], x: \"x\", y: \"pdf\", fill: \"#915F6D\", opacity: 1,\n title: (d) => `P(X=${d.x}) = ${(d.pdf).toPrecision(4)}`}),\n Plot.line(negbinpdf, {filter: plotpois, x: \"x\", y: \"pdfpois\", stroke: \"black\", strokeWidth: 4}),\n Plot.dot(negbinpdf, {filter: plotpois, x: \"x\", y: \"pdfpois\", stroke: \"black\", strokeWidth: 4})\n ]\n })","pinned":false,"hidden":false,"mode":"ojs"},{"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"},{"id":221,"value":"#### Imports","pinned":false,"hidden":false,"mode":"md"},{"id":26,"value":"jstat = require('jstat')","pinned":false,"hidden":false,"mode":"ojs"},{"id":764,"value":"import {Plot} from \"@mkfreeman/plot-tooltip\"","pinned":false,"hidden":false,"mode":"ojs"},{"id":223,"value":"#### Data","pinned":false,"hidden":false,"mode":"md"},{"id":801,"value":"r = params[0]","pinned":true,"hidden":false,"mode":"ojs"},{"id":846,"value":"p = {\nvar p;\n if (parametrization == 'standard'){\n p = params[1]\n }\n else if (parametrization == 'mean'){\n p = params[0]/(params[1] + params[0])\n }\n return p;\n}","pinned":false,"hidden":false,"mode":"ojs"},{"id":805,"value":"mean = r*(1-p)/p","pinned":true,"hidden":false,"mode":"ojs"},{"id":807,"value":"variance = r*(1-p)/(p**2)","pinned":true,"hidden":false,"mode":"ojs"},{"id":17,"value":"negbinpdf = {\n const x = d3.range(0, mean + 4*Math.sqrt(variance), 1);\n const data = x.map(x => ({x: x, pdf: jstat.negbin.pdf(x, r, p), pdfpois: jstat.poisson.pdf(x, mean)}));\n return data\n}","pinned":false,"hidden":false,"mode":"ojs"},{"id":19,"value":"negbincdf = jstat.negbin.cdf(params[2], r, p);","pinned":false,"hidden":false,"mode":"ojs"},{"id":229,"value":"#### Style","pinned":false,"hidden":false,"mode":"md"},{"id":202,"value":"","pinned":false,"hidden":false,"mode":"html"},{"id":58,"value":"viewof md_colour = Inputs.color({label: \"Highlight color cells\", value: \"#F5F5DC\"})","pinned":false,"hidden":false,"mode":"ojs"}],"files":"$W31","stdlib":"1","titling":"auto"}} 2f:"$2e:body"