2b:null 2c:["$","title",null,{"children":"Non-central Chi2-distribution | Mattias Villani | Observable"}] 31:[] 2f:{"id":"8077470d9fc1777b","slug":"non-central-chi2-distribution","title":"Non-central Chi2-distribution","thumbnail":"9a365795bfd072b78850e4f5a4efe6af0deb1b271a99a33baef1e33272fb6ec2","access_level":"public","create_time":"$D2023-10-07T22:18:37.925Z","update_time":"$D2023-10-07T23:05:06.606Z","view_time":null,"first_public_time":"$D2023-10-07T22:18:37.925Z","fork":{"id":"67961e538501b542","slug":"chi2-distribution","title":"Chi2-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":754,"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":"Non-central Chi2-distribution","cells":[{"id":38,"value":"# Non-central Chi2-distribution","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined"},{"id":629,"value":"

${tex`X\\sim \\chi^\\prime_k(\\lambda)`}, for ${tex`X>0`}","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined"},{"id":496,"value":"The density function of a ${tex`X\\sim \\chi^\\prime_k(\\lambda)`} random variable is\n\n${tex`f(x)=\\frac{1}{2}e^{-(x+ \\lambda)/2}\\big(\\frac{x}{\\lambda} \\big)^{k/4-1/2}I_{k/2-1}(\\sqrt{\\lambda x})`}","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined"},{"id":118,"value":"viewof params = Inputs.form([\n Inputs.range([1, 20], {value: 2, step: 0.1, label: tex`k`}),\n Inputs.range([1, 20], {value: 1, step: 0.1, label: tex`\\lambda`}),\n Inputs.range([Number.EPSILON, 20], {value: 2, step: 0.01, label: \"quantile:\"})\n ])","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":461,"value":"dist_quantile = tex`\\text{If } X \\sim \\chi^\\prime_{${k}}(${lambda}) \\text{ then } `","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":555,"value":"moments = tex`\n\\begin{aligned}\n&E( X) =k + \\lambda = ${(k + lambda).toPrecision(2)} \\\\[1em]\n&S( X) =\\sqrt{2(k+2\\lambda)}= ${(Math.sqrt(2*(k+2*lambda))).toPrecision(2)} \\\\[1em]\n&P( X\\leq ${params[2]}) =${(nccdf(params[2], k, lambda)).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: true\n },\n x: {\n label: \"x\",\n axis: true \n },\n y: {\n label: \"f(x)\",\n //axis: false,\n //domain: [0,(k+lambda) + 4*Math.sqrt(2*(k+2*lambda))]\n },\n marks: [\n Plot.ruleY([0]),\n Plot.ruleX([0]),\n Plot.line(pdfdata, {x: \"x\", y: \"pdf\", stroke : \"steelblue\", strokeWidth: 2}),\n Plot.areaY(pdfdata, {filter: d => d.x <= params[2], x: \"x\", y: \"pdf\", fill: \"steelblue\", opacity: 0.2})\n ]\n })","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":610,"value":"## Relations to other distributions","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined"},{"id":625,"value":"Summing *n* squared independent normal variables with unit variance but different means gives a ${tex`\\chi^\\prime_k(\\lambda)`} variable","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined"},{"id":617,"value":">

If ${tex`X_1,X_2,\\ldots,X_{k} \\overset{\\mathrm{indep}}{\\sim} \\operatorname{N}(\\mu_i,1)`} then\n>

${tex`\\sum_{i=1}^{k}X_i^2 \\sim \\chi^\\prime_k(\\lambda)`}

\nwhere ${tex`\\lambda = \\sum_{i=1}^k \\mu_i`}.\n

\n\n ","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined"},{"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","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":690,"value":"besselfunc = require(\"bessel@1.0.2/bessel.js\")","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":223,"value":"#### Data","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined"},{"id":664,"value":"k = params[0]","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":669,"value":"lambda = params[1]","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":700,"value":"binsize = 0.005","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":17,"value":"pdfdata = {\n const x = d3.range(Number.EPSILON, (k+lambda) + 4*Math.sqrt(2*(k+2*lambda)), binsize);\n const pdf = x.map(x => ({x: x, pdf: ncpdf(x, k, lambda)}));\n return pdf\n}","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":692,"value":"#### Functions","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined"},{"id":693,"value":"function ncpdf(x, k, lambda){\n return 0.5*Math.exp(-(x + lambda)/2)*(x/lambda)**(k/4-0.5)*\n besselfunc.besseli(Math.sqrt(lambda*x), k/2-1)\n}","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":19,"value":"nccdf = function(x, k, lambda){\n const xgrid = d3.range(Number.EPSILON, x, binsize);\n return d3.min([binsize*d3.sum(xgrid.map(d => ncpdf(d, k, lambda))),1])\n}","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"