2a:null 2b:["$","title",null,{"children":"Normalfördelningen | Mattias Villani | Observable"}] 31:T55f,plt_pdf_interval = Plot.plot({ color: { legend: true }, x: { label: "x", axis: true, domain: [params[0] - 4*params[1], params[0] + 4*params[1]] }, y: { axis: false, domain: [0,1.01*jstat.normal.pdf(params[0], params[0], params[1])] }, marks: [ Plot.ruleY([0]), Plot.line(normpdf, {x: "x", y: "pdf", stroke : "steelblue", strokeWidth: 2}), // 1 std Plot.areaY(normpdf, {filter: d => d.x >= params[0] - 1*params[1] && d.x <= params[0] + 1*params[1], x: "x", y: "pdf", fill: "#6CB4EE", opacity: 0.5}), // 2 std Plot.areaY(normpdf, {filter: d => d.x > params[0] - 2*params[1] && d.x < params[0] - 1*params[1], x: "x", y: "pdf", fill: "#0066b2", opacity: 0.7}), Plot.areaY(normpdf, {filter: d => d.x >= params[0] + 1*params[1] && d.x <= params[0] + 2*params[1], x: "x", y: "pdf", fill: "#0066b2", opacity: 0.7}), // 3 std Plot.areaY(normpdf, {filter: d => d.x > params[0] - 3*params[1] && d.x <= params[0] - 2*params[1], x: "x", y: "pdf", fill: "#002D62", opacity: 0.7}), Plot.areaY(normpdf, {filter: d => d.x > params[0] + 2*params[1] && d.x <= params[0] + 3*params[1], x: "x", y: "pdf", fill: "#002D62", opacity: 0.7}), Plot.text(textdata, {x: "x", y: "y", text: "text", dy: 0, fontSize: 20, textAnchor: "middle", frameAnchor: "middle"}) ] })32:[] 2e:{"id":"32c30160ed8888fa","slug":"normalfordelningen","title":"Normalfördelningen","thumbnail":"1910bd30862f498bf6715eac5a70d464f95f9fe6817a394b3a74d78f8c08b477","access_level":"public","create_time":"$D2023-01-31T10:53:18.339Z","update_time":"$D2023-09-27T10:55:59.271Z","view_time":null,"first_public_time":"$D2023-01-31T10:53:18.339Z","fork":{"id":"c7c15ac36c4abef7","slug":"student-t","title":"Student-t fördelning (standard version)","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":673,"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":true,"trashed":false,"subscribed":null,"forks":4,"collection":{"id":"209e13808f44ed11","slug":"sda1","title":"SDA1","owner":{"id":"7592a7ab121f74a0","login":"mattiasvillani","name":"Mattias Villani"}},"type":"notebook","body":{"title":"Normalfördelningen","cells":[{"id":38,"value":"# Normalfördelningen","pinned":false,"hidden":false,"mode":"md"},{"id":664,"value":">

${tex`X\\sim \\operatorname{N}(\\mu,\\sigma^2)`}, för ${tex`X \\in (-\\infty,\\infty)`}","pinned":false,"hidden":false,"mode":"md"},{"id":496,"value":"## Sannolikhetsfunktion (täthetsfunktion)","pinned":false,"hidden":false,"mode":"md"},{"id":118,"value":"viewof params = Inputs.form([\n Inputs.range([-3, 3], {value: 0, step: 0.1, label: tex`\\mu:`}),\n Inputs.range([0.1, 3], {value: 1, step: 0.1, label: tex`\\sigma:`}),\n Inputs.range([-10, 10], {value: -1.96, step: 0.01, label: \"Kvantil:\"})\n ])","pinned":false,"hidden":false,"mode":"ojs"},{"id":461,"value":"dist_quantile = tex`\\text{Om } X \\sim \\operatorname{N}(${params[0]},${params[1]}) \\text{ så gäller att }`","pinned":false,"hidden":false,"mode":"ojs"},{"id":529,"value":"moments = tex`\n\\begin{aligned}\n&E( X) = \\mu = ${params[0].toPrecision(3)} \\\\[1em]\n&Var( X) =\\sigma^2 = ${(params[1]**2).toPrecision(3)} \\\\[1em]\n&P( X\\leq ${params[2]}) =${normcdf.toPrecision(4)}\n\\end{aligned}\n`","pinned":false,"hidden":false,"mode":"ojs"},{"id":610,"value":"sidebysideplot = html`\n \n \n \n

\n ${plt_pdf}\n

\n ${plt_pdf_interval}\n

\n \n\n

\n`","pinned":false,"hidden":false,"mode":"ojs"},{"id":614,"value":"
","pinned":false,"hidden":false,"mode":"md"},{"id":498,"value":"## Egenskaper","pinned":false,"hidden":false,"mode":"md"},{"id":507,"value":"### Linjärkombinationer","pinned":false,"hidden":false,"mode":"md"},{"id":509,"value":">

Om ${tex`X \\sim \\operatorname{N}(\\mu,\\sigma^2)`} och ${tex`Y = c + aX`} så gäller\n\n>

${tex`Y \\sim \\operatorname{N}(c + a\\mu, a^2\\sigma^2)`}

\n ","pinned":false,"hidden":false,"mode":"md"},{"id":500,"value":"### Summa av normalfördelade slumpvariabler","pinned":false,"hidden":false,"mode":"md"},{"id":56,"value":">

Om ${tex`X \\sim \\operatorname{N}(\\mu_x,\\sigma_x^2)`} och ${tex`Y \\sim \\operatorname{N}(\\mu_y,\\sigma_y^2)`} är två oberoende slumpvariabler\n\n>

${tex`X + Y \\sim \\operatorname{N}(\\mu_x +\\mu_y,\\sigma_x^2 +\\sigma_y^2)`}

\n ","pinned":false,"hidden":false,"mode":"md"},{"id":417,"value":"Det intressanta med resultatet ovan är **inte** att ${tex`E(X+Y)=\\mu_x +\\mu_y`} eller att ${tex`Var(X+Y)=\\sigma_x^2 +\\sigma_y^2`}, det gäller ju för alla oberoende slumpvariabler. Det intressanta är att **fördelningen** för ${tex`X+Y`} är normalfördelad. *Summan av oberoende normalfördelade variabler är normalfördelad*. Liknande resultat gäller oftast inte för andra fördelningar.","pinned":false,"hidden":false,"mode":"md"},{"id":517,"value":"Resultat ovan kan generaliseras till **beroende** normalfördelade variabler.","pinned":false,"hidden":false,"mode":"md"},{"id":521,"value":">

Om ${tex`X \\sim \\operatorname{N}(\\mu_x,\\sigma_x^2)`} och ${tex`Y \\sim \\operatorname{N}(\\mu_y,\\sigma_y^2)`} är två slumpvariabler\n\n>

${tex`X + Y \\sim \\operatorname{N}(\\mu_x +\\mu_y,\\sigma_x^2 +\\sigma_y^2+ 2 \\rho_{xy}\\sigma_x\\sigma_y)`}

\n \n\n> där ${tex`\\rho_{xy}`} är korrelationen mellan ${tex`X`} och ${tex`Y`}.","pinned":false,"hidden":false,"mode":"md"},{"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":536,"value":"import {interval} from '@mootari/range-slider@1781'","pinned":true,"hidden":false,"mode":"ojs"},{"id":223,"value":"#### Data","pinned":false,"hidden":false,"mode":"md"},{"id":17,"value":"normpdf = {\n const x = d3.range(-10, 10, 0.01);\n const normpdf = x.map(x => ({x: x, pdf: jstat.normal.pdf(x, params[0], params[1])}));\n return normpdf\n}","pinned":false,"hidden":false,"mode":"ojs"},{"id":19,"value":"normcdf = jstat.normal.cdf(params[2], params[0], params[1]);","pinned":false,"hidden":false,"mode":"ojs"},{"id":621,"value":"maxpdf = jstat.normal.pdf(params[0], params[0], params[1])","pinned":false,"hidden":false,"mode":"ojs"},{"id":619,"value":"textdata = [{x: params[0], y: 0.12*maxpdf, text: \"68%\"},{x: params[0] - 1.5*params[1], y: 0.12*maxpdf, text: \"95%\"},{x: params[0] - 2.5*params[1], y: 0.12*maxpdf, text: \"99.7%\"}]","pinned":false,"hidden":false,"mode":"ojs"},{"id":667,"value":"#### Graphs","pinned":false,"hidden":false,"mode":"md"},{"id":546,"value":"$31","pinned":false,"hidden":false,"mode":"ojs"},{"id":305,"value":"plt_pdf = Plot.plot({\n color: {\n legend: true\n },\n x: {\n label: \"x\",\n axis: true\n },\n y: {\n axis: false,\n domain: [0,jstat.normal.pdf(params[0], params[0], params[1])]\n },\n marks: [\n Plot.ruleY([0]),\n Plot.line(normpdf, {x: \"x\", y: \"pdf\", stroke : \"steelblue\", strokeWidth: 2}),\n Plot.areaY(normpdf, {filter: d => d.x <= params[2], x: \"x\", y: \"pdf\", fill: \"steelblue\", opacity: 0.2})\n ]\n })","pinned":false,"hidden":false,"mode":"ojs"},{"id":229,"value":"#### Style","pinned":false,"hidden":false,"mode":"md"},{"id":202,"value":"","pinned":false,"hidden":false,"mode":"html"},{"id":58,"value":"viewof md_colour = Inputs.color({label: \"Highlight color cells\", value: \"#F5F5DC\"})","pinned":false,"hidden":false,"mode":"ojs"}],"files":"$W32","stdlib":"1","titling":"auto"}} 2f:"$2e:body"