2a:null 2b:["$","title",null,{"children":"Chebyshev's inequality | Mattias Villani | Observable"}] 31:Ta5d,viewof params = { var form; if (dist_type == 'normal'){ form = Inputs.form([ Inputs.range([-5, 5], {value: 0, step: 0.1, label: tex`\text{location }\mu`}), Inputs.range([0.1, 5], {value: 1, step: 0.1, label: tex`\text{scale }\sigma`}), ]) } if (dist_type == 'binomial'){ form = Inputs.form([ Inputs.range([1, 30], {value: 10, step: 1, label: tex`\text{number of trials }n`}), Inputs.range([Number.EPSILON, 1 - Number.EPSILON], {value: 0.3, step: 0.01, label: tex`\text{success prob }p`}) ]) } if (dist_type == 'poisson'){ form = Inputs.form([ Inputs.range([0, 10], {value: 1, step: 0.1, label: tex`\text{mean }\lambda`}) ]) } if (dist_type == 'beta'){ form = Inputs.form([ Inputs.range([0.1, 20], {value: 3, step: 0.1, label: tex`\text{shape }\alpha`}), Inputs.range([0.1, 20], {value: 5, step: 0.1, label: tex`\text{shape }\beta`}) ]) } else if (dist_type == 'uniform'){ form = Inputs.form([ Inputs.range([-5, 5], {value: 0, step: 0.1, label: tex`\text{lower limit }a`}), Inputs.range([-5, 5], {value: 1, step: 0.1, label: tex`\text{upper limit }b`}) ]) } else if (dist_type == 'studentt'){ form = Inputs.form([ Inputs.range([-5, 5], {value: 0, step: 0.1, label: tex`\text{location }\mu`}), Inputs.range([0.1, 5], {value: 1, step: 0.1, label: tex`\text{scale }\sigma`}), Inputs.range([3, 50], {value: 4, step: 1, label: tex`\text{degrees of freedom }\nu`}) ]) } else if (dist_type == 'chi2'){ form = Inputs.form([ Inputs.range([1, 50], {value: 3, step: 1, label: tex`\text{degrees of freedom }\nu`}) ]) } else if (dist_type == 'lognormal'){ form = Inputs.form([ Inputs.range([-5,5], {value: 0, step: 0.1, label: tex`\text{location }\mu`}), Inputs.range([0.1,5], {value: 1, step: 0.1, label: tex`\text{scale }\sigma`}) ]) } else if (dist_type == 'bimodal'){ form = Inputs.form([ Inputs.range([-5,5], {value: -2, step: 0.1, label: tex`\text{location }\mu_1`}), Inputs.range([-5,5], {value: 2, step: 0.1, label: tex`\text{location }\mu_2`}), Inputs.range([0.1,5], {value: 1, step: 0.1, label: tex`\text{scale }\sigma_1`}), Inputs.range([0.1,5], {value: 1, step: 0.1, label: tex`\text{scale }\sigma_2`}), Inputs.range([0,1], {value: 0.5, step: 0.01, label: tex`\text{weight }w_1`}) ]) } else if (dist_type == 'bimodal'){ form = 1 } else if (dist_type == 'three-point'){ form = Inputs.form([ Inputs.range([0,1], {value: 1 - 1/k**2, step: 0.01, label: tex`\text{prob at zero}`}) ]) } return form; }32:T8d4,pdfdata = { const ngrid = 500 const lowerprob = 0.001 const upperprob = 0.999 if (dist_type == 'normal'){ const mu = params[0]; const sigma = params[1]; const xgrid = d3.range(jStat.normal.inv(lowerprob, mu, sigma), jStat.normal.inv(upperprob, mu, sigma ), 6*sigma/ngrid) return xgrid.map(d => ({x: d, pdf: jStat.normal.pdf(d, mu, sigma)})) } if (dist_type == 'binomial'){ const n = params[0]; const p = params[1]; const xgrid = d3.range(0,n+1) return xgrid.map(d => ({x: d, pdf: jStat.binomial.pdf(d, n, p)})) } if (dist_type == 'poisson'){ const lambda = params[0]; const xgrid = d3.range(0, 4*Math.sqrt(lambda)) return xgrid.map(d => ({x: d, pdf: jStat.poisson.pdf(d, lambda)})) } if (dist_type == 'beta'){ const alpha = params[0]; const beta = params[1]; const xgrid = d3.range(0, 1, 1/ngrid) return xgrid.map(d => ({x: d, pdf: jStat.beta.pdf(d, alpha, beta)})) } else if (dist_type == 'uniform'){ const min = params[0]; const max = params[1]; const xgrid = d3.range(0, 1, (max-min)/ngrid) return xgrid.map(d => ({x: d, pdf: jStat.uniform.pdf(d, min, max)})) } else if (dist_type == 'studentt'){ const location = params[0]; const scale = params[1]; const df = params[2]; const xgrid = d3.range(location - 4*scale*3, location + 4*scale*3, 2*4*scale*3/ngrid) return xgrid.map(d => ({x: d, pdf: studenttpdf(d, location, scale, df)})) } else if (dist_type == 'chi2'){ const df = params[0]; const lowerlimit = 0; const upperlimit = jStat.chisquare.inv(upperprob, df ) const xgrid = d3.range(lowerlimit, upperlimit, (upperlimit - lowerlimit)/ngrid) return xgrid.map(d => ({x: d, pdf: jStat.chisquare.pdf(d, df)})) } else if (dist_type == 'lognormal'){ const mu = params[0]; const sigma = params[1]; const lowerlimit = 0; const upperlimit = jStat.lognormal.inv(upperprob, mu, sigma); const xgrid = d3.range(lowerlimit, upperlimit, (upperlimit - lowerlimit)/ngrid) return xgrid.map(d => ({x: d, pdf: jStat.lognormal.pdf(d, mu, sigma)})) } else if (dist_type == 'three-point'){ const p = params[0] return [{x: -1, pdf: (1-p)/2},{x: 0, pdf: p},{x: 1, pdf: (1-p)/2}] } }33:T60a,meanstd = { if (dist_type == 'normal'){ const mu = params[0]; const sigma = params[1]; return {mean: mu, std: sigma} } if (dist_type == 'binomial'){ const n = params[0]; const p = params[1]; return {mean: n*p, std: Math.sqrt(n*p*(1-p))} } if (dist_type == 'poisson'){ const lambda = params[0]; return {mean: lambda, std: Math.sqrt(lambda)} } if (dist_type == 'beta'){ const alpha = params[0]; const beta = params[1]; return {mean: jStat.beta.mean(alpha,beta), std: Math.sqrt(jStat.beta.variance(alpha,beta))} } else if (dist_type == 'uniform'){ const min = params[0]; const max = params[1]; return {mean: jStat.uniform.mean(min,max), std: Math.sqrt(jStat.uniform.variance(min,max))} } else if (dist_type == 'studentt'){ const location = params[0]; const scale = params[1]; const df = params[2]; return {mean: location, std: scale*Math.sqrt(df/(df-2))} } else if (dist_type == 'chi2'){ const df = params[0]; const lowerlimit = 0; const upperlimit = jStat.chisquare.inv(0.995, df ) return {mean: df, std: Math.sqrt(2*df)} } else if (dist_type == 'lognormal'){ const mu = params[0]; const sigma = params[1]; const lowerlimit = 0; const upperlimit = jStat.lognormal.inv(0.995, mu, sigma); return {mean: jStat.lognormal.mean(mu,sigma), std: Math.sqrt(jStat.lognormal.variance(mu,sigma))} } else if (dist_type == 'three-point'){ const p = params[0] return {mean: 0, std: Math.sqrt(1-p)} } }34:T765,prob_outside = { if (dist_type == 'normal'){ const mu = params[0]; const sigma = params[1]; return jStat.normal.cdf(meanstd.mean - k*meanstd.std, mu, sigma) + 1-jStat.normal.cdf(meanstd.mean + k*meanstd.std, mu, sigma) } if (dist_type == 'binomial'){ const n = params[0]; const p = params[1]; return jStat.binomial.cdf(meanstd.mean - k*meanstd.std, n, p) + 1-jStat.binomial.cdf(meanstd.mean + k*meanstd.std, n, p) } if (dist_type == 'poisson'){ const lambda = params[0]; return jStat.poisson.cdf(meanstd.mean - k*meanstd.std, lambda) + 1-jStat.poisson.cdf(meanstd.mean + k*meanstd.std, lambda) } if (dist_type == 'beta'){ const alpha = params[0]; const beta = params[1]; return jStat.beta.cdf(meanstd.mean - k*meanstd.std, alpha, beta) + 1-jStat.beta.cdf(meanstd.mean + k*meanstd.std, alpha, beta) } else if (dist_type == 'uniform'){ const min = params[0]; const max = params[1]; return jStat.uniform.cdf(meanstd.mean - k*meanstd.std, min, max) + 1-jStat.uniform.cdf(meanstd.mean + k*meanstd.std, min, max) } else if (dist_type == 'studentt'){ const location = params[0]; const scale = params[1]; const df = params[2]; return studenttcdf(meanstd.mean - k*meanstd.std, location, scale, df) + 1-studenttcdf(meanstd.mean + k*meanstd.std, location, scale, df) } else if (dist_type == 'chi2'){ const df = params[0]; return jStat.chisquare.cdf(meanstd.mean - k*meanstd.std, df) + 1-jStat.chisquare.cdf(meanstd.mean + k*meanstd.std, df) } else if (dist_type == 'lognormal'){ const mu = params[0]; const sigma = params[1]; return jStat.lognormal.cdf(meanstd.mean - k*meanstd.std, mu, sigma) + 1-jStat.lognormal.cdf(meanstd.mean + k*meanstd.std, mu, sigma) } if (dist_type == 'three-point'){ const p = params[0]; return k <= 1/Math.sqrt(1-p) ? 1-p : 0 } }35:[] 2e:{"id":"20856782195d9fb4","slug":"chebyshevs-inequality","title":"Chebyshev's inequality","thumbnail":"28468b43b1b2d9311b990f116f0ecdd084b0691c8c453dbab111793d5080b6d0","access_level":"public","create_time":"$D2025-03-09T16:43:47.615Z","update_time":"$D2025-04-11T09:01:44.750Z","view_time":null,"first_public_time":"$D2025-03-09T18:01:06.620Z","fork":{"id":"82a1b81f0bb3c541","slug":"central-limit-theorem","title":"Central limit theorem","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":1471,"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"},{"type":"team","id":"7592a7ab121f74a0","login":"mattiasvillani","name":"Mattias Villani","avatar_url":"https://avatars-next.observableusercontent.com/a440aaf39a3abefb86cbcc758fd7a5ab42555ae68d6825b52ec006f385a94d1d","tier":"pro_2024","role":"viewer"}],"comments":[],"custom_thumbnail":false,"trashed":false,"subscribed":null,"forks":2,"collection":{"id":"3247547e8812c274","slug":"statistics","title":"Basic Statistics","owner":{"id":"7592a7ab121f74a0","login":"mattiasvillani","name":"Mattias Villani"}},"type":"notebook","body":{"title":"Chebyshev's inequality","cells":[{"id":0,"value":"# Chebyshev's inequality\n","pinned":false,"hidden":false,"mode":"md"},{"id":1445,"value":"> [Chebyshev's inequality](https://en.wikipedia.org/wiki/Chebyshev%27s_inequality) gives an upper bound on the probability that a random variable deviates a certain number of standard deviations from its mean. \n\nSince the inequality applies to *all* random variables (with finite variance) regardless of distributional shape, the inequality is typically far from sharp. That is, the actual probability is for most distributions substantially smaller than the Chebyshev upper bound, as you can see below in the interactive widget. The real use of the inequality is for mathematical proofs.\n\nA lower bound that covers all distributions cannot be found however, since it is known that a certain three-point distribution has a probability that attains the upper bound for all ${tex`k>1`}. This is the 'three-point' distribution in the drop-down menu below.","pinned":false,"hidden":false,"mode":"md"},{"id":925,"value":">

\nLet ${tex`X`} be a random variable with mean ${tex`\\mu`} and variance ${tex`\\sigma^2`}.
\nThen, for any constant ${tex`k>0`}, \n${tex.block`\\mathrm{Pr}\\big(\\vert X - \\mu \\vert \\geq k\\sigma\\big) \\leq \\frac{1}{k^2}`}","pinned":false,"hidden":false,"mode":"md"},{"id":729,"value":"## Interactive exploration","pinned":false,"hidden":false,"mode":"md"},{"id":287,"value":"viewof dist_type = Inputs.select(['beta', 'binomial', 'chi2', 'lognormal', 'normal', 'poisson', 'studentt', 'three-point', 'uniform',], {value: \"normal\", label: \"distribution:\"})","pinned":false,"hidden":false,"mode":"ojs"},{"id":792,"value":"$31","pinned":false,"hidden":false,"mode":"ojs"},{"id":1364,"value":"viewof k = Inputs.range([1.01, 10], {label: tex`\\text{number of stdev, }k`, value: 2, step: 0.01})","pinned":false,"hidden":false,"mode":"ojs"},{"id":1385,"value":"**True probability**: ${tex`\\hspace{0.8cm}\\mathrm{Pr}\\big(\\vert X - \\mu \\vert \\geq ${k}\\sigma\\big) = ${(prob_outside).toFixed(5)}`}
\n**Chebychev's bound**: ${tex`\\hspace{0.4cm}\\mathrm{Pr}\\big(\\vert X - \\mu \\vert \\geq ${k}\\sigma\\big) \\leq \\frac{1}{${k}^2} = ${(1/k**2).toFixed(5)}`} ","pinned":false,"hidden":false,"mode":"md"},{"id":1336,"value":"pltdata = {\n if (discrete){\n return Plot.plot({\n marks: [\n Plot.barY(pdfdata, {x: \"x\", y: \"pdf\", fill: \"steelblue\", strokeWidth: 0, opacity: 0.8}),\n Plot.ruleY([0])\n ]\n }) \n }\n else {\n return Plot.plot({\n marks: [\n Plot.line(pdfdata, {x: \"x\", y: \"pdf\", stroke : \"steelblue\", strokeWidth: 2}),\n Plot.areaY(pdfdata, {filter: d => d.x <= meanstd.mean - k*meanstd.std, x: \"x\", y: \"pdf\", fill: \"steelblue\", opacity: 0.2}),\n Plot.areaY(pdfdata, {filter: d => d.x >= meanstd.mean + k*meanstd.std, x: \"x\", y: \"pdf\", fill: \"steelblue\", opacity: 0.2}),\n Plot.ruleY([0])\n ]\n })\n }\n}","pinned":false,"hidden":false,"mode":"ojs"},{"id":1013,"value":"
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\n\n---\n## 🙈 Appendix 🙈\n\nJust some code and implementation details that you probably don't want to read.","pinned":false,"hidden":false,"mode":"md"},{"id":765,"value":"#### Imports","pinned":false,"hidden":false,"mode":"md"},{"id":770,"value":"jStat = require('jstat')","pinned":false,"hidden":false,"mode":"ojs"},{"id":62,"value":"d3 = require(\"https://d3js.org/d3.v5.min.js\")\n","pinned":false,"hidden":false,"mode":"ojs"},{"id":1078,"value":"#### Style","pinned":false,"hidden":false,"mode":"md"},{"id":874,"value":"html``","pinned":false,"hidden":false,"mode":"ojs"},{"id":1029,"value":"viewof md_colour = Inputs.color({label: \"Highlight color cells\", value: \"#F5F5DC\"})","pinned":false,"hidden":false,"mode":"ojs"},{"id":760,"value":"#### Data","pinned":false,"hidden":false,"mode":"md"},{"id":685,"value":"params_ = params","pinned":false,"hidden":false,"mode":"ojs"},{"id":27,"value":"$32","pinned":false,"hidden":false,"mode":"ojs"},{"id":1372,"value":"$33","pinned":false,"hidden":false,"mode":"ojs"},{"id":1380,"value":"$34","pinned":false,"hidden":false,"mode":"ojs"},{"id":572,"value":"#### Code","pinned":false,"hidden":false,"mode":"md"},{"id":1245,"value":"discrete = ([\"poisson\", \"binomial\", \"three-point\"].includes(dist_type))","pinned":false,"hidden":false,"mode":"ojs"},{"id":1345,"value":"function studenttpdf(x, mu, sigma, df){\n return jStat.studentt.pdf((x-mu)/sigma, df)/sigma\n}","pinned":false,"hidden":false,"mode":"ojs"},{"id":1348,"value":"function studenttcdf(x, mu, sigma, df){\n return jStat.studentt.cdf((x-mu)/sigma, df)\n}","pinned":false,"hidden":false,"mode":"ojs"},{"id":79,"value":"\nheight = 250\n","pinned":false,"hidden":false,"mode":"ojs"},{"id":86,"value":"width = 800","pinned":false,"hidden":false,"mode":"ojs"},{"id":72,"value":"margin = ({top: 20, right: 20, bottom: 30, left: 40})","pinned":false,"hidden":false,"mode":"ojs"},{"id":1193,"value":"#### Detached plots","pinned":false,"hidden":false,"mode":"md"}],"files":"$W35","stdlib":"1","titling":"auto"}} 2f:"$2e:body"