2a:null 2b:["$","title",null,{"children":"Approximating the Binomial distribution | Mattias Villani | Observable"}] 31:T577,cdfs = { var cdf; if (normalapprox){ if (poisapprox){ cdf = tex` \begin{array}{ll} \text{Exact:} & P( X\leq ${kvantil}) =${binomcdf.toPrecision(4)} \\[0.25em] \text{Normal approx:} & P( X\leq ${kvantil}) =${normalapproxcdf.toPrecision(4)} \\[0.25em] \text{Normal approx with correction:} & P( X\leq ${kvantil}) =${normalapproxcdf_corrected.toPrecision(4)} \\[0.25em] \text{Poisson approx} & P( X\leq ${kvantil}) =${poissonapproxcdf.toPrecision(4)} \end{array} ` } else{ cdf = tex` \begin{array}{ll} \text{Exact:} & P( X\leq ${kvantil}) =${binomcdf.toPrecision(4)} \\[0.25em] \text{Normal approx:} & P( X\leq ${kvantil}) =${normalapproxcdf.toPrecision(4)} \\[0.25em] \text{Normal approx with correction:} & P( X\leq ${kvantil}) =${normalapproxcdf_corrected.toPrecision(4)} \end{array} ` } } else{ if (poisapprox){ cdf = tex` \begin{array}{ll} \text{Exact:} & P( X\leq ${kvantil}) =${binomcdf.toPrecision(4)} \\[0.25em] \text{Poisson approx} & P( X\leq ${kvantil}) =${poissonapproxcdf.toPrecision(4)} \end{array} ` } else{ cdf = tex` \begin{array}{ll} \text{Exact:} & P( X\leq ${kvantil}) =${binomcdf.toPrecision(4)} \end{array} ` } } return cdf; }32:T649,plt_pdf = Plot.plot({ color: { legend: true, domain: legenddomain.filter( (d,i) => boolselect[i]), range: legendcolor.filter( (d,i) => boolselect[i]) }, x: { label: "x", axis: true, domain: xrange }, y: { label: "f(x)", domain: [0,1.1*jstat.binomial.pdf(params[0]*params[1], params[0],params[1])] }, tooltip: { fill: "b39640", stroke: "b39640", opacity: 1, }, marks: [ Plot.ruleY([0]), Plot.barY(approxpdf,{x: "x", y: "binompdf", fill : "#C1CFE4", strokeWidth: 0, opacity: 0.8, title: (d) => `P(X=${d.x}) = ${(d.binompdf).toPrecision(4)}`}), Plot.barY(approxpdf, {filter: d => d.x <= kvantil, x: "x", y: "binompdf", fill: "#98afd2", opacity: 1, title: (d) => `P(X=${d.x}) = ${(d.binompdf).toPrecision(4)}`}), Plot.line(approxpdf, {filter: d => normalapprox, x: "x", y: "normalpdf", stroke : "#AB8D61", strokeWidth: 3}), Plot.dot(approxpdf, {filter: d => normalapprox, x: "x", y: "normalpdf", stroke : "#AB8D61", strokeWidth: 3}), Plot.line(approxpdf, {filter: d => normalapproxcorr, x: "x", y: "normalpdfcorr", stroke : "#007878", strokeWidth: 3}), Plot.dot(approxpdf, {filter: d => normalapproxcorr, x: "x", y: "normalpdfcorr", stroke : "#007878", strokeWidth: 3}), Plot.ruleX(approxpdf, {filter: d => poisapprox && d.x >= 0, x: "x", y: "poissonpdf", stroke: "#780000", strokeWidth: 2}), Plot.dot(approxpdf, {filter: d => poisapprox && d.x >= 0, x: "x", y: "poissonpdf", fill: "#780000", r: 5}), ] })33:[] 2e:{"id":"31eff05135f2cd87","slug":"approximating-the-binomial-distribution","title":"Approximating the Binomial distribution","thumbnail":"22435d5247bac50ec60cf0b544544f9fbe362f9813643cbe46cff9b95553af22","access_level":"public","create_time":"$D2024-03-01T10:52:44.079Z","update_time":"$D2024-05-26T08:08:19.162Z","view_time":null,"first_public_time":"$D2024-03-01T12:58:23.070Z","fork":{"id":"8e97614a2bda7e05","slug":"binomial-distribution","title":"Binomial 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":1143,"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"},{"type":"team","id":"7592a7ab121f74a0","login":"mattiasvillani","name":"Mattias Villani","avatar_url":"https://avatars-next.observableusercontent.com/a440aaf39a3abefb86cbcc758fd7a5ab42555ae68d6825b52ec006f385a94d1d","tier":"pro_2024","role":"editor"}],"comments":[],"custom_thumbnail":false,"trashed":false,"subscribed":null,"forks":0,"collection":{"id":"05cb69ead7642f7e","slug":"probability","title":"Probability","owner":{"id":"7592a7ab121f74a0","login":"mattiasvillani","name":"Mattias Villani"}},"type":"notebook","body":{"title":"Approximating the Binomial distribution","cells":[{"id":38,"value":"# Approximating the Binomial distribution","pinned":false,"hidden":false,"mode":"md"},{"id":1078,"value":"## Approximations\nThe **Binomial distribution** ${tex`X \\sim \\mathrm{Bin}(n,p)`} can be approximated by:\n\n- **Normal distribution** ${tex.block`N\\big(np,\\sqrt{np(1-p)}\\big)`} A rule of thumb: accurate when ${tex`np>10`} and ${tex`n(1-p)>10`}.\n- **Poisson distribution** ${tex.block`\\mathrm{Poisson}(\\lambda = np)`} which is an accurate approximation when *n* is large and *p* is small.
A rule of thumb: accurate when ${tex`n>20`} and ${tex`p<0.05`}.","pinned":false,"hidden":false,"mode":"md"},{"id":496,"value":"## Interactive exploration","pinned":false,"hidden":false,"mode":"md"},{"id":118,"value":"viewof params = Inputs.form([\n Inputs.range([1, 200], {value: 10, step: 1, label: tex`n`}),\n Inputs.range([0.001, 1], {value: 0.5, step: 0.001, label: tex`p`}),\n ])","pinned":false,"hidden":false,"mode":"ojs"},{"id":665,"value":"viewof kvantil = Inputs.range([0, params[0]], {value: 3, step: 1, label: \"quantile\"})","pinned":false,"hidden":false,"mode":"ojs"},{"id":555,"value":"moments = tex`\n\\text{If } X \\sim \\operatorname{Binom}(${params[0]},${params[1]}) \\text{ then }\\\\[0.5em]\n\n\nE( X) =n p = ${(params[0]*params[1]).toPrecision(3)} \\\\[0.5em]\nVar( X) =n p (1-p) =${(params[0]*params[1]*(1-params[1])).toPrecision(3)} \n\n`","pinned":false,"hidden":false,"mode":"ojs"},{"id":722,"value":"$31","pinned":false,"hidden":false,"mode":"ojs"},{"id":1034,"value":"viewof plotselector = Inputs.checkbox([\"normal\", \"normal with correction\", \"poisson\"], {label: \"plot approximation:\", value: [\"normal approximation\", \"normal approximation corrected\", \"poisson approximation\"]})","pinned":false,"hidden":false,"mode":"ojs"},{"id":305,"value":"$32","pinned":false,"hidden":false,"mode":"ojs"},{"id":109,"value":"
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\n\n---\n## 🙈 Appendix 🙈\n\nHär följer bara kod och detaljer som du antagligen inte vill läsa.","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":697,"value":"import {Plot} from \"@mkfreeman/plot-tooltip\"","pinned":false,"hidden":false,"mode":"ojs"},{"id":223,"value":"#### Data","pinned":false,"hidden":false,"mode":"md"},{"id":1019,"value":"normalapprox = plotselector.includes(\"normal\")","pinned":false,"hidden":false,"mode":"ojs"},{"id":1053,"value":"normalapproxcorr = plotselector.includes(\"normal with correction\")","pinned":false,"hidden":false,"mode":"ojs"},{"id":1022,"value":"poisapprox = plotselector.includes(\"poisson\")","pinned":false,"hidden":false,"mode":"ojs"},{"id":17,"value":"binompdf = {\n const x = xrange;\n const binompdf = x.map(x => ({x: x, pdf: jstat.binomial.pdf(x, params[0], params[1]), type: \"binomial\"}));\n const normalapproxpdf = x.map(x => ({x: x, pdf: jstat.normal.pdf(x, params[0]*params[1], Math.sqrt(params[0]*params[1]*(1-params[1]))),type: \"normal\"}));\n const data = binompdf.concat(normalapproxpdf)\n return data\n}","pinned":false,"hidden":false,"mode":"ojs"},{"id":605,"value":"approxpdf = {\n const x = xrange;\n const approxpdf = x.map(x => ({\n x: x, \n binompdf: jstat.binomial.pdf(x, params[0], params[1]),\n normalpdf: jstat.normal.pdf(x, params[0]*params[1], Math.sqrt(params[0]*params[1]*(1-params[1]))), \n normalpdfcorr: jstat.normal.pdf(x + 0.5, params[0]*params[1], Math.sqrt(params[0]*params[1]*(1-params[1]))),\n poissonpdf: jstat.poisson.pdf(x, params[0]*params[1])\n }));\n return approxpdf\n}","pinned":false,"hidden":false,"mode":"ojs"},{"id":1003,"value":" high = Math.ceil(jstat.normal.inv(0.9999, params[0]*params[1], Math.sqrt(params[0]*params[1]*(1-params[1]))))","pinned":true,"hidden":false,"mode":"ojs"},{"id":1001,"value":"low = Math.floor(jstat.normal.inv(0.0001, params[0]*params[1], Math.sqrt(params[0]*params[1]*(1-params[1]))))","pinned":true,"hidden":false,"mode":"ojs"},{"id":1006,"value":"xrange = d3.range(low, high + 1, 1);","pinned":false,"hidden":false,"mode":"ojs"},{"id":19,"value":"binomcdf = jstat.binomial.cdf(kvantil, params[0], params[1]);","pinned":false,"hidden":false,"mode":"ojs"},{"id":717,"value":"normalapproxcdf = jstat.normal.cdf(kvantil, params[0]*params[1], Math.sqrt(params[0]*params[1]*(1-params[1])))\n ","pinned":false,"hidden":false,"mode":"ojs"},{"id":753,"value":"normalapproxcdf_corrected = jstat.normal.cdf(kvantil+0.5, params[0]*params[1], Math.sqrt(params[0]*params[1]*(1-params[1])))\n","pinned":false,"hidden":false,"mode":"ojs"},{"id":942,"value":"poissonapproxcdf = jstat.poisson.cdf(kvantil, params[0]*params[1])","pinned":false,"hidden":false,"mode":"ojs"},{"id":1104,"value":"legenddomain = [\"binomial\", \"normal\", \"normal with correction\", \"poisson\"]","pinned":true,"hidden":false,"mode":"ojs"},{"id":1106,"value":"legendcolor = [\"#C1CFE4\", \"#AB8D61\", \"#007878\", \"#780000\"]","pinned":true,"hidden":false,"mode":"ojs"},{"id":1108,"value":"boolselect = [true,normalapprox, normalapproxcorr, poisapprox]","pinned":true,"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":"$W33","stdlib":"1","titling":"auto"}} 2f:"$2e:body"