2a:null 2b:["$","title",null,{"children":"Normal (Gaussian) distribution | Mattias Villani | Observable"}] 31:T55f,plt_pdf_interval = Plot.plot({ color: { legend: true }, x: { label: "x", axis: true, domain: [params[0] - 4*params[1], params[0] + 4*params[1]] }, y: { axis: false, domain: [0,1.01*jstat.normal.pdf(params[0], params[0], params[1])] }, marks: [ Plot.ruleY([0]), Plot.line(normpdf, {x: "x", y: "pdf", stroke : "steelblue", strokeWidth: 2}), // 1 std Plot.areaY(normpdf, {filter: d => d.x >= params[0] - 1*params[1] && d.x <= params[0] + 1*params[1], x: "x", y: "pdf", fill: "#6CB4EE", opacity: 0.5}), // 2 std Plot.areaY(normpdf, {filter: d => d.x > params[0] - 2*params[1] && d.x < params[0] - 1*params[1], x: "x", y: "pdf", fill: "#0066b2", opacity: 0.7}), Plot.areaY(normpdf, {filter: d => d.x >= params[0] + 1*params[1] && d.x <= params[0] + 2*params[1], x: "x", y: "pdf", fill: "#0066b2", opacity: 0.7}), // 3 std Plot.areaY(normpdf, {filter: d => d.x > params[0] - 3*params[1] && d.x <= params[0] - 2*params[1], x: "x", y: "pdf", fill: "#002D62", opacity: 0.7}), Plot.areaY(normpdf, {filter: d => d.x > params[0] + 2*params[1] && d.x <= params[0] + 3*params[1], x: "x", y: "pdf", fill: "#002D62", opacity: 0.7}), Plot.text(textdata, {x: "x", y: "y", text: "text", dy: 0, fontSize: 20, textAnchor: "middle", frameAnchor: "middle"}) ] })32:[] 2e:{"id":"632a4c506a5ed159","slug":"normal-gaussian-distribution","title":"Normal (Gaussian) distribution","thumbnail":"553b398d87378cf35f850f5e39d74108d29588262ec9b7a7a6dc7dd3af57a2dc","access_level":"public","create_time":"$D2023-02-25T15:05:07.014Z","update_time":"$D2024-02-16T16:47:53.498Z","view_time":null,"first_public_time":"$D2023-02-25T15:05:07.014Z","fork":{"id":"32c30160ed8888fa","slug":"normalfordelningen","title":"Normalfördelningen","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":705,"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":1,"collection":{"id":"40c8ef5af1a51aff","slug":"distributions","title":"Statistical distributions","owner":{"id":"7592a7ab121f74a0","login":"mattiasvillani","name":"Mattias Villani"}},"type":"notebook","body":{"title":"Normal (Gaussian) distribution","cells":[{"id":38,"value":"# Normal (Gaussian) distribution","pinned":false,"hidden":false,"mode":"md"},{"id":664,"value":"

${tex`X\\sim \\operatorname{N}(\\mu,\\sigma^2)`}, for ${tex`X \\in (-\\infty,\\infty)`}","pinned":false,"hidden":false,"mode":"md"},{"id":496,"value":"The density function of a ${tex`X\\sim N(\\mu,\\sigma^2)`} random variable is\n\n${tex`f(x)=\\frac{1}{\\sqrt{2\\pi\\sigma^2}}e^{-\\frac{1}{2\\sigma^2}(x-\\mu)^2}`}","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: \"quantile:\"})\n ])","pinned":false,"hidden":false,"mode":"ojs"},{"id":461,"value":"dist_quantile = tex`\\text{If } X \\sim \\operatorname{N}(${params[0]},${params[1]}) \\text{ then }`","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":"## Properties","pinned":false,"hidden":false,"mode":"md"},{"id":507,"value":"### Linear combinations","pinned":false,"hidden":false,"mode":"md"},{"id":509,"value":">

If ${tex`X \\sim \\operatorname{N}(\\mu,\\sigma^2)`} and ${tex`Y = c + aX`} then\n\n>

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

\n ","pinned":false,"hidden":false,"mode":"md"},{"id":500,"value":"### Sum of normally distributed variables","pinned":false,"hidden":false,"mode":"md"},{"id":56,"value":">

If ${tex`X \\sim \\operatorname{N}(\\mu_x,\\sigma_x^2)`} and ${tex`Y \\sim \\operatorname{N}(\\mu_y,\\sigma_y^2)`} are two independent variables, then\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":"The interesting aspect of the above result is **not** that ${tex`E(X+Y)=\\mu_x +\\mu_y`} or that ${tex`Var(X+Y)=\\sigma_x^2 +\\sigma_y^2`}, since this holds for all independent variables. The interesting result here is that the **distribution** of the sum ${tex`X+Y`} is also normally distributed. Similar results are not in general true for other distributions.","pinned":false,"hidden":false,"mode":"md"},{"id":517,"value":"The result above can be generalized to **dependent** normally distributed variables.","pinned":false,"hidden":false,"mode":"md"},{"id":521,"value":">

If ${tex`X \\sim \\operatorname{N}(\\mu_x,\\sigma_x^2)`} and ${tex`Y \\sim \\operatorname{N}(\\mu_y,\\sigma_y^2)`}, then\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> where ${tex`\\rho_{xy}`} is the correlation between ${tex`X`} och ${tex`Y`}.","pinned":false,"hidden":false,"mode":"md"},{"id":109,"value":"
\n
\n
\n
\n
\n
\n
\n
\n
\n
\n
\n
\n
\n
\n
\n
\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":681,"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"