2b:null 2c:["$","title",null,{"children":"Simulations | abhisek banerjee | Observable"}] 31:T62c,svgPMF = { const width = 600, height = 300; const margin = { top: 20, right: 30, bottom: 30, left: 40 }; const svg = d3.create("svg") .attr("width", width) .attr("height", height); // Count the occurrences of each unique value in the samples const counts = samples.reduce((acc, val) => { acc[val] = (acc[val] || 0) + 1; return acc; }, {}); // Calculate the probabilities for each unique value const probabilities = Object.entries(counts).map(([value, count]) => ({ value: +value, probability: count / samples.length })); const x = d3.scaleBand() .domain(probabilities.map(d => d.value)) .range([margin.left, width - margin.right]) .padding(0.1); const y = d3.scaleLinear() .domain([0, d3.max(probabilities, d => d.probability)]).nice() .range([height - margin.bottom, margin.top]); svg.append("g") .selectAll("rect") .data(probabilities) .join("rect") .attr("x", d => x(d.value)) .attr("y", d => y(d.probability)) .attr("width", x.bandwidth()) .attr("height", d => y(0) - y(d.probability)) .attr("fill", "steelblue"); svg.append("g") .attr("transform", `translate(0,${height - margin.bottom})`) .call(d3.axisBottom(x)); svg.append("g") .attr("transform", `translate(${margin.left},0)`) .call(d3.axisLeft(y)); svg.append("text") .attr("x", width / 2) .attr("y", margin.top / 2) .attr("text-anchor", "middle") .style("font-size", "16px") .text(`PMF Visualization for ${discrete_distribution} Distribution`); return svg.node(); }32:T5f4,svgCDF = { const width = 600, height = 300; const margin = { top: 20, right: 30, bottom: 30, left: 40 }; const svg = d3.create("svg") .attr("width", width) .attr("height", height); // Sort the samples and calculate the cumulative probabilities const sortedSamples = samples.slice().sort((a, b) => a - b); const cdfData = []; let cumulative = 0; sortedSamples.forEach((value, index) => { cumulative += 1 / samples.length; if (index === 0 || value !== sortedSamples[index - 1]) { cdfData.push({ value, cumulative }); } }); const x = d3.scaleBand() .domain(cdfData.map(d => d.value)) .range([margin.left, width - margin.right]); const y = d3.scaleLinear() .domain([0, 1]) .range([height - margin.bottom, margin.top]); const line = d3.line() .x(d => x(d.value) + x.bandwidth() / 2) .y(d => y(d.cumulative)) .curve(d3.curveStepAfter); svg.append("path") .datum(cdfData) .attr("fill", "none") .attr("stroke", "steelblue") .attr("stroke-width", 1.5) .attr("d", line); svg.append("g") .attr("transform", `translate(0,${height - margin.bottom})`) .call(d3.axisBottom(x)); svg.append("g") .attr("transform", `translate(${margin.left},0)`) .call(d3.axisLeft(y)); svg.append("text") .attr("x", width / 2) .attr("y", margin.top / 2) .attr("text-anchor", "middle") .style("font-size", "16px") .text(`CDF Visualization for ${discrete_distribution} Distribution`); return svg.node(); }33:T576,svgPDFcont = { const width = 600, height = 300; const margin = { top: 20, right: 30, bottom: 30, left: 40 }; // Create a histogram from the samples to approximate the PDF const histogram = d3.bin() .domain(d3.extent(cont_samples)) .thresholds(30)(cont_samples); const x = d3.scaleLinear() .domain(d3.extent(cont_samples)) .range([margin.left, width - margin.right]); const y = d3.scaleLinear() .domain([0, d3.max(histogram, d => d.length / cont_samples.length)]) .range([height - margin.bottom, margin.top]); const svg = d3.create("svg") .attr("width", width) .attr("height", height); svg.append("g") .selectAll("rect") .data(histogram) .join("rect") .attr("x", d => x(d.x0)) .attr("y", d => y(d.length / cont_samples.length)) .attr("width", d => x(d.x1) - x(d.x0) - 1) .attr("height", d => y(0) - y(d.length / cont_samples.length)) .attr("fill", "steelblue"); svg.append("g") .attr("transform", `translate(0,${height - margin.bottom})`) .call(d3.axisBottom(x)); svg.append("g") .attr("transform", `translate(${margin.left},0)`) .call(d3.axisLeft(y)); svg.append("text") .attr("x", width / 2) .attr("y", margin.top / 2) .attr("text-anchor", "middle") .style("font-size", "16px") .text(`PDF Approximation for ${cont_distribution} Distribution`); return svg.node(); }34:T541,svgCDFcont = { const width = 600, height = 300; const margin = { top: 20, right: 30, bottom: 30, left: 40 }; // Sort samples and calculate cumulative values const sortedSamples = cont_samples.slice().sort(d3.ascending); const cdfData = sortedSamples.map((d, i) => ({ value: d, cumulative: (i + 1) / sortedSamples.length })); const x = d3.scaleLinear() .domain(d3.extent(sortedSamples)) .range([margin.left, width - margin.right]); const y = d3.scaleLinear() .domain([0, 1]) .range([height - margin.bottom, margin.top]); const svg = d3.create("svg") .attr("width", width) .attr("height", height); // Draw the CDF line const line = d3.line() .x(d => x(d.value)) .y(d => y(d.cumulative)); svg.append("path") .datum(cdfData) .attr("fill", "none") .attr("stroke", "steelblue") .attr("stroke-width", 1.5) .attr("d", line); svg.append("g") .attr("transform", `translate(0,${height - margin.bottom})`) .call(d3.axisBottom(x)); svg.append("g") .attr("transform", `translate(${margin.left},0)`) .call(d3.axisLeft(y)); svg.append("text") .attr("x", width / 2) .attr("y", margin.top / 2) .attr("text-anchor", "middle") .style("font-size", "16px") .text(`CDF for ${cont_distribution} Distribution`); return svg.node(); }35:T4c7,svgRiemannSum = { const width = 600, height = 300; const margin = { top: 20, right: 30, bottom: 30, left: 40 }; const svg = d3.create("svg") .attr("width", width) .attr("height", height); const x = d3.scaleLinear() .domain([0, 1]) .range([margin.left, width - margin.right]); const y = d3.scaleLinear() .domain([0, 1]) .range([height - margin.bottom, margin.top]); // Draw the rectangles representing Riemann sums for (let xi = 0; xi < 1; xi += riemannStepSize) { svg.append("rect") .attr("x", x(xi)) .attr("y", y(f(xi))) .attr("width", x(xi + riemannStepSize) - x(xi)) .attr("height", y(0) - y(f(xi))) .attr("fill", "steelblue") .attr("opacity", 0.5); } // Draw the axes svg.append("g") .attr("transform", `translate(0,${height - margin.bottom})`) .call(d3.axisBottom(x)); svg.append("g") .attr("transform", `translate(${margin.left},0)`) .call(d3.axisLeft(y)); svg.append("text") .attr("x", width / 2) .attr("y", margin.top / 2) .attr("text-anchor", "middle") .style("font-size", "16px") .text(`Riemann Sum Approximation of ∫x² dx with Δx = ${riemannStepSize}`); return svg.node(); }36:T521,svgLebesgueSum = { const width = 600, height = 300; const margin = { top: 20, right: 30, bottom: 30, left: 40 }; const svg = d3.create("svg") .attr("width", width) .attr("height", height); const x = d3.scaleLinear() .domain([0, 1]) .range([margin.left, width - margin.right]); const y = d3.scaleLinear() .domain([0, 1]) .range([height - margin.bottom, margin.top]); // Draw rectangles representing the Lebesgue sums for (let yi = 0; yi < f(1); yi += lebesgueStepSize) { const xStart = Math.sqrt(yi); const xEnd = Math.sqrt(yi + lebesgueStepSize); svg.append("rect") .attr("x", x(xStart)) .attr("y", y(yi)) .attr("width", x(xEnd) - x(xStart)) .attr("height", y(0) - y(lebesgueStepSize)) .attr("fill", "orange") .attr("opacity", 0.5); } // Draw the axes svg.append("g") .attr("transform", `translate(0,${height - margin.bottom})`) .call(d3.axisBottom(x)); svg.append("g") .attr("transform", `translate(${margin.left},0)`) .call(d3.axisLeft(y)); svg.append("text") .attr("x", width / 2) .attr("y", margin.top / 2) .attr("text-anchor", "middle") .style("font-size", "16px") .text(`Lebesgue Sum Approximation of ∫x² dx with Δy = ${lebesgueStepSize}`); return svg.node(); }37:T7a1,function generateContinuousSamples(cont_distribution, sampleSize) { switch (cont_distribution) { case "Normal": if (typeof mean !== 'undefined' && typeof stdDev !== 'undefined') { // Using d3's randomNormal to generate samples const normal = d3.randomNormal(mean, stdDev); return Array.from({ length: sampleSize1 }, () => normal()); } break; case "Uniform": // Generate uniform samples in the range [0, 1) const uniform = d3.randomUniform(0, 1); return Array.from({ length: sampleSize1 }, () => uniform()); case "Exponential": if (typeof lambda !== 'undefined') { // Using d3's randomExponential to generate samples const exponential = d3.randomExponential(1/lambdacont); return Array.from({ length: sampleSize1 }, () => exponential()); } break; case "Beta": if (typeof alpha !== 'undefined' && typeof beta !== 'undefined') { return Array.from({ length: sampleSize1 }, () => jStat.beta.sample(alpha, beta)); } break; case "Gamma": if (typeof alpha !== 'undefined') { // Gamma with shape parameter `alpha` and scale set to 1 return Array.from({ length: sampleSize1 }, () => jStat.gamma.sample(alpha, theta)); } break; case "Lognormal": if (typeof mean !== 'undefined' && typeof stdDev !== 'undefined') { // Using d3's randomLogNormal to generate samples const logNormal = d3.randomLogNormal(mean, stdDev); return Array.from({ length: sampleSize1 }, () => logNormal()); } break; case "Chi-Square": if (typeof alpha !== 'undefined') { // Using jStat's chi-square distribution return Array.from({ length: sampleSize1 }, () => jStat.chisquare.sample(alpha,1)); } break; default: return []; } return []; // Return an empty array if the required parameters are not available } 38:T76f,function plotHistogramWithDensity(data, title, densityFunction, densityParams) { const width = 600, height = 300; const margin = { top: 20, right: 30, bottom: 30, left: 40 }; const svg = d3.create("svg") .attr("width", width) .attr("height", height); // Create histogram bins const histogram = d3.bin() .domain(d3.extent(data)) .thresholds(30)(data); const x = d3.scaleLinear() .domain(d3.extent(data)) .range([margin.left, width - margin.right]); const y = d3.scaleLinear() .domain([0, d3.max(histogram, d => d.length / data.length)]) .range([height - margin.bottom, margin.top]); // Draw histogram bars svg.append("g") .selectAll("rect") .data(histogram) .join("rect") .attr("x", d => x(d.x0)) .attr("y", d => y(d.length / data.length)) .attr("width", d => x(d.x1) - x(d.x0) - 1) .attr("height", d => y(0) - y(d.length / data.length)) .attr("fill", "steelblue"); // Generate points for the density line const densityLineData = d3.range(x.domain()[0], x.domain()[1], (x.domain()[1] - x.domain()[0]) / 100) .map(xValue => ({ x: xValue, y: densityFunction(xValue, ...densityParams) })); // Draw the density curve const line = d3.line() .x(d => x(d.x)) .y(d => y(d.y)) .curve(d3.curveBasis); svg.append("path") .datum(densityLineData) .attr("fill", "none") .attr("stroke", "red") .attr("stroke-width", 2) .attr("d", line); // Draw axes svg.append("g") .attr("transform", `translate(0,${height - margin.bottom})`) .call(d3.axisBottom(x)); svg.append("g") .attr("transform", `translate(${margin.left},0)`) .call(d3.axisLeft(y)); // Add title svg.append("text") .attr("x", width / 2) .attr("y", margin.top / 2) .attr("text-anchor", "middle") .style("font-size", "16px") .text(title); return svg.node(); }39:T150f,infinitesimal_flow_slider = { // Observable-style slider (Inputs.range) controlling "time" t const W = 920, H = 420, pad = 16; // --- Controls const nInput = Inputs.range([10, 10000], { label: "Sample Size", step: 10, value: 1000 }); const tInput = Inputs.range([0, 1], { label: "Flow time t", step: 0.001, value: 0.0 }); const speedInput = Inputs.range([0.0, 2.5], { label: "Auto-play speed (0 = off)", step: 0.05, value: 0.0 }); const ui = html`

${nInput}

${tInput}

${speedInput}

`; // --- SVG const svg = d3.create("svg") .attr("viewBox", `0 0 ${W} ${H}`) .style("width", "100%") .style("max-width", `${W}px`) .style("height", `${H}px`) .style("border", "1px solid rgba(255,255,255,0.18)") .style("border-radius", "12px"); svg.append("rect") .attr("x", 0).attr("y", 0) .attr("width", W).attr("height", H) .attr("fill", "rgba(255,255,255,0.92)"); svg.append("text") .attr("x", 14).attr("y", 28) .attr("font-size", 16) .attr("font-weight", 700) .attr("fill", "rgba(0,0,0,0.82)") .text("Infinitesimal Flow: points evolve continuously with time t"); svg.append("text") .attr("x", 14).attr("y", 52) .attr("font-size", 12) .attr("fill", "rgba(0,0,0,0.55)") .text("ODE: dz/dt = v_t(z). Drag t to morph the distribution."); const plot = svg.append("g").attr("transform", `translate(0,0)`); // --- Latent space scales const xS = d3.scaleLinear().domain([-4, 4]).range([pad, W - pad]); const yS = d3.scaleLinear().domain([-3.5, 3.5]).range([H - pad, 70]); // subtle grid const grid = plot.append("g").attr("opacity", 0.18); for (let gx = -4; gx <= 4; gx += 1) { grid.append("line") .attr("x1", xS(gx)).attr("x2", xS(gx)) .attr("y1", yS(-3.5)).attr("y2", yS(3.5)) .attr("stroke", "black"); } for (let gy = -3; gy <= 3; gy += 1) { grid.append("line") .attr("x1", xS(-4)).attr("x2", xS(4)) .attr("y1", yS(gy)).attr("y2", yS(gy)) .attr("stroke", "black"); } // --- Random helpers (Box–Muller) function randn() { let u = 0, v = 0; while (u === 0) u = Math.random(); while (v === 0) v = Math.random(); return Math.sqrt(-2 * Math.log(u)) * Math.cos(2 * Math.PI * v); } // --- Time-dependent velocity field v_t(z) function vField(t, x, y) { // Moving center (creates evolving swirl) const cx = 0.9 * Math.sin(2 * Math.PI * t); const cy = 0.6 * Math.cos(2 * Math.PI * t); const dx = x - cx, dy = y - cy; // Components: swirl + squeeze + mild bend (toy CNF) const swirl = 1.05; const squeeze = 0.65 * Math.sin(2 * Math.PI * t); const bend = 0.35 * Math.tanh(x * y); const vx = -swirl * dy + squeeze * x + bend; const vy = swirl * dx - squeeze * y - 0.25 * Math.tanh(x); return [vx, vy]; } // --- RK2 integrator from 0 to tEnd (fast enough for interactive use) function rk2Step(t, x, y, dt) { const [k1x, k1y] = vField(t, x, y); const xm = x + 0.5 * dt * k1x; const ym = y + 0.5 * dt * k1y; const [k2x, k2y] = vField(t + 0.5 * dt, xm, ym); return [x + dt * k2x, y + dt * k2y]; } function evolve(points0, tEnd) { const dt = 0.01; // stable + quick const steps = Math.max(1, Math.ceil(tEnd / dt)); const h = tEnd / steps; let pts = points0.map(p => p.slice()); let t = 0; for (let s = 0; s < steps; s++) { for (let i = 0; i < pts.length; i++) { const [x, y] = pts[i]; pts[i] = rk2Step(t, x, y, h); } t += h; } return pts; } // --- State let pts0 = []; let circles = plot.append("g").selectAll("circle"); function resample(n) { pts0 = d3.range(n).map(() => [randn(), 0.85 * randn()]); } function draw(n, t) { if (pts0.length !== n) resample(n); const ptsT = (t === 0) ? pts0 : evolve(pts0, t); circles = circles .data(ptsT, (d, i) => i) .join( enter => enter.append("circle") .attr("r", 1.35) .attr("opacity", 0.75) .attr("cx", d => xS(d[0])) .attr("cy", d => yS(d[1])), update => update, exit => exit.remove() ) .attr("cx", d => xS(d[0])) .attr("cy", d => yS(d[1])); } // initial render draw(nInput.value, tInput.value); // --- Interactivity nInput.addEventListener("input", () => { // resample when sample size changes resample(nInput.value); draw(nInput.value, tInput.value); }); tInput.addEventListener("input", () => { draw(nInput.value, tInput.value); }); // --- Auto-play (optional) let last = null; function tick(ts) { if (last === null) last = ts; const dtms = ts - last; last = ts; const speed = speedInput.value; if (speed > 0) { let t = tInput.value + (speed * dtms / 1000); if (t > 1) t = t - 1; // wrap tInput.value = t; draw(nInput.value, t); } requestAnimationFrame(tick); } requestAnimationFrame(tick); // Return combined view (controls + viz) return html`

${ui} ${svg.node()}

`; }3a:[] 2f:{"id":"4a47a63aafed1ef8","slug":null,"title":"Simulations","thumbnail":"cb4a7e609c311e820db6d631ac075ba6237e06b5b123d4bc5a973cd80d608769","access_level":"public","create_time":"$D2024-09-30T01:12:40.167Z","update_time":"$D2026-02-02T17:55:31.105Z","view_time":null,"first_public_time":"$D2024-09-30T05:59:58.108Z","has_unpublished_edits":null,"fork":{"id":"7dcef0ebff4028d9","slug":"input-chart","title":"Input + Chart","owner":{"id":"f35c755083683fe5","login":"observablehq","name":"Observable"}},"stars":0,"starred":false,"owner":{"type":"team","id":"ae09c890574f6c53","login":"abhisek-banerjee","name":"abhisek banerjee","avatar_url":"https://avatars-next.observableusercontent.com/d5466158559b26b1e5d19aab8c889889deb937452d84c49dfd6ccae261a0884b","tier":"starter_2024"},"version":581,"publish_version":null,"role":null,"authored":false,"authors":[{"type":"individual","id":"26c7ddda3d62c7d0","login":"abhisek-banerjee","name":"abhisek banerjee","avatar_url":"https://avatars-next.observableusercontent.com/d5466158559b26b1e5d19aab8c889889deb937452d84c49dfd6ccae261a0884b","tier":null}],"members":[{"type":"individual","id":"26c7ddda3d62c7d0","login":"abhisek-banerjee","name":"abhisek banerjee","avatar_url":"https://avatars-next.observableusercontent.com/d5466158559b26b1e5d19aab8c889889deb937452d84c49dfd6ccae261a0884b","tier":null,"role":"editor"},{"type":"team","id":"ae09c890574f6c53","login":"abhisek-banerjee","name":"abhisek banerjee","avatar_url":"https://avatars-next.observableusercontent.com/d5466158559b26b1e5d19aab8c889889deb937452d84c49dfd6ccae261a0884b","tier":"starter_2024","role":"editor"}],"comments":[],"custom_thumbnail":false,"trashed":false,"lapsed":null,"subscribed":null,"forks":0,"collection":null,"type":"notebook","body":{"title":"Simulations","cells":[{"id":4,"value":"# Simulations\n### Abhisek Banerjee\n","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","output":"","database":"$undefined"},{"id":28,"value":"\nWe have two types of distributions:\n- Conteneous Distributions\n- Discrete Distributions\n","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","output":"","database":"$undefined"},{"id":31,"value":"## Simulating Discrete Distributions\n","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","output":"","database":"$undefined"},{"id":33,"value":"Here, we simulate the probability mass function (PMF) and the cumulative distribution functions of some basic discrete distributions such as:\n- Binomial distribution\n- Poisson distribution\n- Geometric distribution","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","output":"","database":"$undefined"},{"id":40,"value":"First we will select which discrete distribution we want..","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","output":"","database":"$undefined"},{"id":42,"value":"viewof discrete_distribution = Inputs.select(\n [\"Binomial\", \"Poisson\", \"Geometric\"],\n { label: \"Select Discrete Distribution\" }\n);","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":97,"value":"### Now we plot Probability Mass Function (PMF)","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","output":"","database":"$undefined"},{"id":99,"value":"$31","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":104,"value":"// Create an interactive slider for selecting the sample size\nviewof sampleSize = Inputs.range([10, 10000], { \n label: \"Sample Size\", \n step: 10, \n value: 1000 \n});","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":144,"value":"viewof trialsInput = (discrete_distribution === \"Binomial\" || discrete_distribution === \"Negative Binomial\") \n ? Inputs.range([1, 100], { label: \"Number of Trials\", step: 1, value: 10 }) \n : Inputs.range([1, 1], { disabled: true }); // Disabled if not applicable\n","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":148,"value":"viewof probSuccessInput = [\"Binomial\", \"Geometric\", \"Negative Binomial\"].includes(discrete_distribution) \n ? Inputs.range([0.01, 1], { label: \"Probability of Success\", step: 0.01, value: 0.5 }) \n : Inputs.range([1, 1], { disabled: true }); // Disabled if not applicable\n","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":152,"value":"viewof lambdaInput = (discrete_distribution === \"Poisson\") \n ? Inputs.range([0.1, 10], { label: \"Lambda (Rate of Events)\", step: 0.1, value: 3 }) \n : Inputs.range([0, 0], { disabled: true }); // Disabled if not applicable","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":155,"value":"### Now we plot Cumulative distribiution functions (CDF)","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","output":"","database":"$undefined"},{"id":158,"value":"$32","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":165,"value":"---","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","output":"","database":"$undefined"},{"id":167,"value":"## Simulating Conteneous Distributions","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","output":"","database":"$undefined"},{"id":171,"value":"Here, we simulate the probability desity function (PDF) and the cumulative distribution functions (CDF) of some basic discrete distributions such as:\n- Normal distribution\n- Uniform distribution\n- Exponential distribution\n- Beta distribution\n- Gamma distribution\n- Lognormal distribution\n- Chi-Square distribution\n","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","output":"","database":"$undefined"},{"id":176,"value":"First we will select which conteneous distribution we want..","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","output":"","database":"$undefined"},{"id":178,"value":"viewof cont_distribution = Inputs.select(\n [\"Normal\", \"Uniform\", \"Exponential\", \"Beta\", \"Gamma\", \"Lognormal\", \"Chi-Square\"],\n { label: \"Select Contineous Distribution\" }\n);","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":222,"value":"$33","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":247,"value":"// Create an interactive slider for selecting the sample size\nviewof sampleSize1 = Inputs.range([10, 10000], { \n label: \"Sample Size\", \n step: 10, \n value: 1000 \n});","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":227,"value":"viewof mean = [\"Normal\", \"Lognormal\"].includes(cont_distribution) \n ? Inputs.range([-10, 10], { label: \"Mean\", step: 0.1, value: 0 }) \n : Inputs.range([1, 1], { disabled: true }); // Disabled if not applicable","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":235,"value":"viewof stdDev = [\"Normal\", \"Lognormal\"].includes(cont_distribution) \n ? Inputs.range([0.1, 10], { label: \"Standard Deviation\", step: 0.1, value: 1 }) \n : Inputs.range([1, 1], { disabled: true }); // Disabled if not applicable\n","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":231,"value":"viewof alpha = [\"Beta\", \"Gamma\",\"Chi-Square\"].includes(cont_distribution) \n ? Inputs.range([0.1, 10], { label: \"Alpha (Shape Parameter)\", step: 0.1, value: 2 }) \n : Inputs.range([1, 1], { disabled: true }); // Disabled if not applicable\n","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":250,"value":"viewof theta = (cont_distribution === \"Gamma\") \n ? Inputs.range([0.1, 10], { label: \"Scale Parameter (Theta)\", step: 0.1, value: 1 }) \n : Inputs.range([1, 1], { disabled: true }); // Disabled if not applicable;","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":239,"value":"viewof beta = (cont_distribution === \"Beta\") \n ? Inputs.range([0.1, 10], { label: \"Beta (Shape Parameter)\", step: 0.1, value: 5 }) \n : Inputs.range([1, 1], { disabled: true }); // Disabled if not applicable\n","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":243,"value":"viewof lambdacont = (cont_distribution === \"Exponential\") \n ? Inputs.range([0.1, 10], { label: \"Lambda (Rate)\", step: 0.1, value: 1 }) \n : Inputs.range([1, 1], { disabled: true }); // Disabled if not applicable\n","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":278,"value":"### Now we plot Cumulative distribiution functions (CDF)","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","output":"","database":"$undefined"},{"id":280,"value":"$34","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":360,"value":"---","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","output":"","database":"$undefined"},{"id":363,"value":"## Transformation of Random Variables","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","output":"","database":"$undefined"},{"id":365,"value":"### 1. Exponential to Gamma distribution","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","output":"","database":"$undefined"},{"id":367,"value":"If a set of random variables follow exponential distribution with parameter (say: lambda = 1), then the sum of such (k) random variables will follow a gamma distribution with parameter (k,1).\n- We simulate the sum of k random variables following Exp(1) distribution.\n- We fit histogram of the sum of the random variables\n- We pass an exponential density curve to the histogram for comparision.","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","output":"","database":"$undefined"},{"id":321,"value":"viewof kGamma = Inputs.range([10, 30], { label: \"Number of Exponentials to Sum (k)\", step: 1, value: 15 });\n","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":380,"value":"viewof sampleSize3 = Inputs.range([100, 5000], { label: \"Sample Size\", step: 100, value: 1000 });","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":383,"value":"svgGamma2 = plotHistogramWithDensity(\n samplesGammaFromExponential, \n `Gamma Distribution from Sum of ${kGamma} Exponentials`, \n gammaPDF, \n [kGamma, 1] // k, theta (where theta = 1)\n);","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":402,"value":"### 2. Chi-Sq distribution from sum of squared standard normal distributions","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","output":"","database":"$undefined"},{"id":407,"value":"We take a couple of standard normal distribution. Let's say 'n' of them. We, the, square them and add them up. The resultant random variable will follow a chi square distribution with 'n' degrees of freedom. ","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","output":"","database":"$undefined"},{"id":461,"value":"viewof sampleSize4 = Inputs.range([100, 5000], { label: \"Sample Size\", step: 100, value: 1000 });","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":332,"value":"viewof kChiSquare = Inputs.range([1, 10], { label: \"Degrees of Freedom (k) for Chi-Square\", step: 1, value: 3 });\n","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":409,"value":"svgChiSquare = plotHistogramWithDensity(\n samplesChiSquareFromNormal, \n `Chi-Square Distribution from ${kChiSquare} Squared Normals`, \n chiSquarePDF, \n [kChiSquare] // Degrees of freedom (k)\n);","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":437,"value":"### 3. Normal to log-normal distribution","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","output":"","database":"$undefined"},{"id":443,"value":"The Lognormal distribution is derived from a Normal distribution by applying the exponential function. Specifically, if a random variable X is normally distributed, then Y = e^X follows a Lognormal distribution. This transformation is often used in finance (e.g., modeling stock prices) and in natural processes (e.g., distribution of biological populations).","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","output":"","database":"$undefined"},{"id":445,"value":"viewof normalMean = Inputs.range([-2, 2], { label: \"Mean of Normal Distribution\", step: 0.1, value: 0 });\n","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":448,"value":"viewof normalStdDev = Inputs.range([0.1, 2], { label: \"Standard Deviation of Normal Distribution\", step: 0.1, value: 1 });","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":457,"value":"viewof sampleSize5 = Inputs.range([100, 10000], { label: \"Sample Size\", step: 1, value: 1000 });","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":464,"value":"svgLognormal = plotHistogramWithDensity(\n samplesLognormalFromNormal, \n `Lognormal Distribution from Normal with Mean = ${normalMean} and Std Dev = ${normalStdDev}`, \n lognormalPDF, \n [normalMean, normalStdDev] // Mean and standard deviation of the underlying Normal distribution\n);","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":471,"value":"---","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","output":"","database":"$undefined"},{"id":473,"value":"## Riemann Integration vs Lebesgue Integration","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","output":"","database":"$undefined"},{"id":500,"value":"To illustrate the difference between Lebesgue integration and Riemann integration, we demonstrated how these methods approximate an integral in fundamentally different ways. This can be visualized through the construction of Lebesgue sums and Riemann sums for a given function.\n- **Riemann Integration:** Divides the domain of the function (i.e., the x-axis) into subintervals and sums up the areas of rectangles under the curve.\n- **Lebesgue Integration:** Divides the range of the function (i.e., the y-axis) into levels, measures the x-axis areas that contribute to each range, and sums up the products of these areas and the corresponding function values.\n\nBoth approaches aim to calculate the same result—the integral—but they approach the task in different ways, which makes each suitable for different types of functions.","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","output":"","database":"$undefined"},{"id":525,"value":"\n

Riemann Integration

Riemann integration works by dividing the domain of the function into small intervals and summing the areas of rectangles that approximate the area under the curve. This is known as the Riemann sum.

The Riemann sum is defined as:

\n${tex.block`S_R = \\sum_{i=1}^{n} f(x_i^*) \\delta x\")`}

where:

\n- [a,b] is the interval of integration\n- delta represents the width of each subinterval.\n- (x_i^*) is the value of the function at some point within the interval.\n\n

The Riemann approach is straightforward for continuous functions, where the value of the function doesn't change drastically over the interval. It works by essentially stacking up rectangles under the function curve.

\n`","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","output":"","database":"$undefined"},{"id":490,"value":"$35","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":550,"value":"viewof riemannStepSize = Inputs.range([0.01, 0.1], { label: \"Riemann Step Size (Δx)\", step: 0.01, value: 0.05 });\n","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":563,"value":"riemannSum = {\n let sum = 0;\n for (let x = 0; x < 1; x += riemannStepSize) {\n const height = f(x);\n sum += height * riemannStepSize;\n }\n return sum;\n}","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":542,"value":"

Lebesgue Integration

Lebesgue integration takes a different approach by focusing on the range of the function rather than the domain. Instead of summing over small intervals of the x-axis, we partition the y-axis (the range of values the function can take) into small levels.

The Lebesgue integral is computed by adding up slices of the range and their corresponding measure of the domain:

${tex.block`S_L = \\sum_{j=1}^{m} y_j \\mu(A_j)\")`}

where:

\n\n- $y_j are levels that partition the range of the function.\n- A_j is the preimage of y_j, i.e., all x such that [y_j, y_j + \\\\Delta y]\n- \\mu(A_j) represents the measure (or width) of the corresponding set A_j.\n\n

This approach allows Lebesgue integration to effectively handle complex functions with many discontinuities, as the focus is on how often different values of the function occur rather than on partitioning the domain.

\n`","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","output":"","database":"$undefined"},{"id":492,"value":"$36","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":547,"value":"viewof lebesgueStepSize = Inputs.range([0.1, 1], { label: \"Lebesgue Level Size (Δy)\", step: 0.1, value: 0.5 });","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":566,"value":"lebesgueSum = {\n let sum = 0;\n for (let y = 0; y < f(1); y += lebesgueStepSize) {\n // Calculate the measure of the pre-image of the interval [y, y + Δy]\n const xStart = Math.sqrt(y);\n const xEnd = Math.sqrt(y + lebesgueStepSize);\n const width = xEnd - xStart;\n sum += y * width;\n }\n return sum;\n}","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":283,"value":"*Code appendix*","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","output":"","database":"$undefined"},{"id":287,"value":"jStat = require(\"jstat@latest\");","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":292,"value":"trials = (discrete_distribution === \"Binomial\" || discrete_distribution === \"Negative Binomial\") ? trialsInput : -1;\n","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":296,"value":"probSuccess = [\"Binomial\", \"Geometric\", \"Negative Binomial\"].includes(discrete_distribution) ? probSuccessInput : 0.5;\n","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":300,"value":"\nlambda = (discrete_distribution === \"Poisson\") ? lambdaInput : -1;","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":304,"value":"function generateDiscreteSamples(discrete_distribution, sampleSize) {\n switch (discrete_distribution) {\n case \"Binomial\":\n if (typeof trials !== 'undefined' && typeof probSuccess !== 'undefined') {\n return Array.from({ length: sampleSize }, () => d3.randomBinomial(trials, probSuccess)(sampleSize));\n }\n break;\n case \"Poisson\":\n if (typeof lambda !== 'undefined') {\n return Array.from({ length: sampleSize }, () => d3.randomPoisson(lambda)(sampleSize));\n }\n break;\n case \"Geometric\":\n if (typeof probSuccess !== 'undefined') {\n return Array.from({ length: sampleSize }, () => d3.randomGeometric(probSuccess)(sampleSize));\n }\n break;\n default:\n return [];\n }\n return []; // Return an empty array if the required parameters are not available\n}\n","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":308,"value":"samples = generateDiscreteSamples(discrete_distribution, sampleSize);","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":312,"value":"$37","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":316,"value":"cont_samples = generateContinuousSamples(cont_distribution, sampleSize);","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":391,"value":"samplesGammaFromExponential = Array.from({ length: sampleSize3 }, () => {\n let sum = 0;\n for (let i = 0; i < kGamma; i++) {\n sum += jStat.exponential.sample(1); // Rate parameter = 1 for each exponential\n }\n return sum;\n});","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":394,"value":"function gammaPDF(x, k, theta) {\n if (x < 0) return 0;\n return (Math.pow(x, k - 1) * Math.exp(-x / theta)) / (Math.pow(1, k) * jStat.gammafn(k));\n}","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":421,"value":"function chiSquarePDF(x, k) {\n if (x < 0) return 0;\n return (Math.pow(x, (k / 2) - 1) * Math.exp(-x / 2)) / (Math.pow(2, k / 2) * jStat.gammafn(k / 2));\n}","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":425,"value":"samplesChiSquareFromNormal = {\n return Array.from({ length: sampleSize4 }, () => {\n let sumOfSquares = 0;\n for (let i = 0; i < kChiSquare; i++) {\n const normalSample = jStat.normal.sample(0, 1);\n sumOfSquares += normalSample ** 2;\n }\n return sumOfSquares;\n });\n}","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":431,"value":"$38","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":451,"value":"samplesLognormalFromNormal = {\n // Generate Normal random samples with selected mean and standard deviation\n const normalSamples = Array.from({ length: sampleSize5 }, () => jStat.normal.sample(normalMean, normalStdDev));\n // Transform them to Lognormal using the exponential function\n return normalSamples.map(x => Math.exp(x));\n}","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":453,"value":"function lognormalPDF(x, mean, stdDev) {\n if (x <= 0) return 0;\n return (1 / (x * stdDev * Math.sqrt(2 * Math.PI))) * Math.exp(-((Math.log(x) - mean) ** 2) / (2 * stdDev ** 2));\n}","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":508,"value":"katex = require('katex@0.11.1/dist/katex.min.js')","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":557,"value":"function f(x) {\n return x ** 2;\n}","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":580,"value":"$39","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"}],"files":"$W3a","stdlib":"1","titling":"auto"}} 30:"$2f:body"