Student-t Distribution

jstat = require('jstat')

normcdf = jstat.normal.cdf(myinputs[1], 0, 1);
studentcdf = jstat.studentt.cdf(myinputs[1], myinputs[0]);

viewof myinputs = Inputs.form([
      Inputs.range([1, 100], {value: 4, step: 1, label: tex`\text{DF }\nu:`}),
      Inputs.range([-8, 8], {value: -1.96, step: 0.01, label: "Quantile:"})
    ])

tailprobs = tex`\text{Normal distribution: } P(X\leq ${myinputs[1]})=${normcdf.toPrecision(4)} \\ \text{Student-}t \text{ distribution: } P(X\leq ${myinputs[1]})=${studentcdf.toPrecision(4)}`

data_t = {
  const x = d3.range(-8, 8, 0.01);
  const normpdf = x.map(x => ({x: x, pdf: jstat.normal.pdf(x, 0, 1), dist: "normal"}));
  const studentpdf = x.map(x => ({x: x, pdf: jstat.studentt.pdf(x, myinputs[0]), dist: "student-t"}));
  const data = normpdf.concat(studentpdf)
  return data
}

plt_t = Plot.plot({
    style: {fontSize: "16px"},
    color: {
      legend: true,
      style: {fontSize: "16px"}
    },
    x: {
      label: "x",
      axis: true
    },
    y: {
      axis: false,
      domain: [0,0.4]
    },
    marks: [
      Plot.ruleY([0]),
      Plot.line(data_t, {x: "x", y: "pdf", stroke : "dist", strokeWidth: 2}),
      Plot.areaY(data_t, {filter: d => d.x <= myinputs[1] && d.dist == "normal", x: "x", y: "pdf", fill: "steelblue", opacity: 0.2}),
      Plot.areaY(data_t, {filter: d => d.x <= myinputs[1] && d.dist == "student-t", x: "x", y: "pdf", fill: "orange", opacity: 0.2}),
    ]
  })

Figure 1: https://observablehq.com/@mattiasvillani/student-t

viewof myinputs_zoom = Inputs.form([
      Inputs.range([1, 100], {value: 10, step: 1, label: tex`\text{DF }\nu:`}),
      Inputs.range([-8, -3], {value: -5, step: 0.01, label: "Quantile:"})
    ])

data_zoom = {
  const x = d3.range(-10, -3, 0.01);
  const normpdf = x.map(x => ({x: x, pdf: jstat.normal.pdf(x, 0, 1), dist: "normal"}));
  const studentpdf = x.map(x => ({x: x, pdf: jstat.studentt.pdf(x, myinputs_zoom[0]), dist: "student-t"}));
  const data = normpdf.concat(studentpdf)
  return data
}

Plot.plot({
    caption: html`Zoom in the left tail`,
    style: {fontSize: "16px"},
    color: {
      legend: true,
      style: {fontSize: "16px"}
    },
    x: {
      label: "x",
      axis: true,
      ticks: [-8,-7,-6,-5,-4,-3, myinputs_zoom[1]],
      domain: [-8,-3]
    },
    y: {
      axis: true,
      domain: [0,0.01]
    },
    marks: [
      Plot.ruleY([0]),     
      Plot.ruleX([-10]), 
      Plot.ruleX([myinputs_zoom[1]], {stroke: "gray", strokeOpacity: 0.4}),
      Plot.line(data_zoom, {filter: d => d.x <= -3, x: "x", y: "pdf", stroke : "dist", strokeWidth: 2}),
      //Plot.areaY(data, {filter: d => d.x <= myinputs[1] && d.dist == "normal", x: "x", y: "pdf", fill: "steelblue", opacity: 0.2}),
      //Plot.areaY(data, {filter: d => d.x <= myinputs[1] && d.dist == "student-t", x: "x", y: "pdf", fill: "orange", opacity: 0.2}),
    ]
  })

Figure 2: https://observablehq.com/@mattiasvillani/student-t