2a:null 2b:["$","title",null,{"children":"Inverse Gaussian (Wald) distribution | Mattias Villani | Observable"}] 31:[] 2e:{"id":"5d58e6a216ca0aed","slug":"inverse-gaussian-distribution","title":"Inverse Gaussian (Wald) distribution","thumbnail":"9b51c3761641d85d2b7e073007db911be0fd08883f23bae85b707c489a9e5895","access_level":"public","create_time":"$D2024-05-09T11:35:06.519Z","update_time":"$D2024-05-26T12:38:45.413Z","view_time":null,"first_public_time":"$D2024-05-09T11:35:06.519Z","fork":{"id":"d668b1efe2f23f3c","slug":"scaled-inverse-chi-2-distribution","title":"Scaled inverse 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":896,"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":"Inverse Gaussian (Wald) distribution","cells":[{"id":38,"value":"# Inverse Gaussian (Wald) distribution","pinned":false,"hidden":false,"mode":"md"},{"id":662,"value":"disttext = md`

${tex`X\\sim \\operatorname{IG}(\\mu,\\lambda)`}, for ${tex`X \\in (0,\\infty)`}`","pinned":false,"hidden":false,"mode":"ojs"},{"id":496,"value":"## Density function and moments","pinned":false,"hidden":false,"mode":"md"},{"id":883,"value":"tex`p(x)= \\sqrt{\\frac{\\lambda}{2\\pi x^3}}\\exp\\Big(-\\frac{\\lambda(x-\\mu)^2}{2\\mu^2 x}\\Big)`","pinned":false,"hidden":false,"mode":"ojs"},{"id":118,"value":"viewof params = Inputs.form([\n Inputs.range([0.1, 100], {value: 3, step: 0.1, label: tex`\\mu`}),\n Inputs.range([0.1, 5], {value: 1, step: 0.1, label: tex`\\lambda`}),\n Inputs.range([0, 10], {value: 1, step: 0.01, label: \"Quantile:\"})\n ])","pinned":false,"hidden":false,"mode":"ojs"},{"id":461,"value":"dist_quantile = tex`\\text{If } X \\sim \\operatorname{IG}(${params[0]},${params[1]}) \\text{ then } `","pinned":false,"hidden":false,"mode":"ojs"},{"id":555,"value":"moments = tex`\n\\begin{aligned}\n&E( X) =\\mu = ${mean.toPrecision(3)} \\\\[1em]\n&S( X) = \\sqrt{\\frac{\\mu^3}{\\lambda}} = ${Math.sqrt(variance).toPrecision(3)} \\\\[1em]\n&P( X\\leq ${quantile}) =${cdfdata.toPrecision(4)}\n\\end{aligned}\n`","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 label: \"f(x)\",\n //axis: false,\n domain: [0, inversegaussianpdf(mu*(Math.sqrt(1 + (9*mu**2)/(4*lambda**2) ) - (3*mu)/(2*lambda)) , mu, lambda) ] \n },\n marks: [\n Plot.ruleY([0]),\n Plot.ruleX([0]),\n Plot.line(pdfdata, {x: \"x\", y: \"pdf\", stroke : \"red\", strokeWidth: 2}),\n Plot.areaY(pdfdata, {filter: d => d.x <= quantile, x: \"x\", y: \"pdf\", fill: \"red\", opacity: 0.2})\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 want to ignore.","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":841,"value":"#### Functions","pinned":false,"hidden":false,"mode":"md"},{"id":839,"value":"function inversegaussianpdf(x, mu, lambda){\n const pdf = Math.sqrt(lambda/(2*Math.PI*x**3))*Math.exp(-(lambda*(x-mu)**2)/((2*mu**2)*x))\n return pdf\n}","pinned":false,"hidden":false,"mode":"ojs"},{"id":850,"value":"inversegaussiancdf = function(x, mu, lambda){\n const z1 = Math.sqrt(lambda/x)*(x/mu - 1)\n const z2 = -Math.sqrt(lambda/x)*(x/mu + 1)\n return jstat.normal.cdf(z1,0,1) + Math.exp(2*lambda/mu)*jstat.normal.cdf(z2,0,1)\n\n}","pinned":false,"hidden":false,"mode":"ojs"},{"id":223,"value":"#### Data","pinned":false,"hidden":false,"mode":"md"},{"id":829,"value":"mu = params[0]","pinned":false,"hidden":false,"mode":"ojs"},{"id":831,"value":"lambda = params[1]","pinned":false,"hidden":false,"mode":"ojs"},{"id":833,"value":"quantile = params[2]","pinned":false,"hidden":false,"mode":"ojs"},{"id":17,"value":"pdfdata = {\n const x = d3.range(0, 10, 0.01);\n const pdf = x.map(x => ({x: x, pdf: inversegaussianpdf(x, mu, lambda)}));\n return pdf\n}","pinned":false,"hidden":false,"mode":"ojs"},{"id":19,"value":"cdfdata = inversegaussiancdf(quantile, mu, lambda)","pinned":false,"hidden":false,"mode":"ojs"},{"id":767,"value":"mean = mu","pinned":false,"hidden":false,"mode":"ojs"},{"id":769,"value":"variance = mu**3/lambda","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"