30:T7a8,md` In probability theory, the central limit theorem (CLT) establishes that, in many situations, when independent random variables are added, their properly normalized sum tends toward a normal distribution (informally a bell curve) even if the original variables themselves are not normally distributed. The theorem is a key concept in probability theory because it implies that probabilistic and statistical methods that work for normal distributions can be applicable to many problems involving other types of distributions. If ${tex.block`X_{1},X_{2},...,X_{n}`} are random samples each of size ${tex `n`}taken from a population with overall mean ${tex `\mu `} and finite variance ${tex `\sigma ^{2}`} and if ${tex` {\bar {X}}`} is the sample mean, the limiting form of the distribution of ${tex.block` Z=\left({\frac {{\bar {X}}_{n}-\mu }{\sigma /\surd n}}\right)`} as ${tex` n\to \infty `}, is the standard normal distribution For example, suppose that a sample is obtained containing many observations, each observation being randomly generated in a way that does not depend on the values of the other observations, and that the arithmetic mean of the observed values is computed. If this procedure is performed many times, the central limit theorem says that the probability distribution of the average will closely approximate a normal distribution. A simple example of this is that if one flips a coin many times, the probability of getting a given number of heads will approach a normal distribution, with the mean equal to half the total number of flips. At the limit of an infinite number of flips, it will equal a normal distribution. The central limit theorem has several variants. In its common form, the random variables must be identically distributed. In variants, convergence of the mean to the normal distribution also occurs for non-identical distributions or for non-independent observations, if they comply with certain conditions. `31:[] 2e:{"title":"Statistical Notes","cells":[{"id":794,"value":"md`# Statistical Notes`","pinned":false,"hidden":false,"mode":"ojs"},{"id":7,"value":"md`## Probability Distribution`","pinned":false,"hidden":false,"mode":"ojs"},{"id":744,"value":"md`A probability distribution is a function that describes the likelihood of obtaining the possible values that a random variable can assume. In other words, the values of the variable vary based on the underlying probability distribution.`","pinned":false,"hidden":false,"mode":"ojs"},{"id":22,"value":"md` \n**Normal Distribution**\n\n${tex.block`N(x|\\mu,\\sigma) \\sim \\frac{1}{\\sigma\\sqrt{2\\pi}} e^{-\\frac{(x-\\mu)^2}{2\\sigma^2}}`}\n`","pinned":false,"hidden":false,"mode":"ojs"},{"id":39,"value":"viewof normParams = form(html`\n\n`)","pinned":false,"hidden":false,"mode":"ojs"},{"id":183,"value":"html`\nμ = ${normParams.mu}
\nσ = ${normParams.sigma}\n`","pinned":false,"hidden":false,"mode":"ojs"},{"id":51,"value":"html`${pltNorm.render('html')}`","pinned":false,"hidden":false,"mode":"ojs"},{"id":159,"value":"md` \n**Exponential Distribution**\n\n${tex.block`f(x|\\lambda) = \\begin{cases} \\lambda e^{-\\lambda x} & x \\ge 0 \\\\ 0 & x < 0 \\end{cases}`}\n`","pinned":false,"hidden":false,"mode":"ojs"},{"id":166,"value":"viewof expParams = form(html`\n\n`)","pinned":false,"hidden":false,"mode":"ojs"},{"id":179,"value":"html`λ = ${expParams.lambda}`","pinned":false,"hidden":false,"mode":"ojs"},{"id":219,"value":"html`${pltExp.render('html')}`","pinned":false,"hidden":false,"mode":"ojs"},{"id":188,"value":"md` \n**Poisson Distribution**\n\n${tex.block`f(x|\\lambda) = P(X=x|\\lambda) = \\begin{cases} \\tfrac{\\lambda^x}{x!}e^{-\\lambda} & \\text{ for } x = 0,1,2,\\ldots \\\\ 0 & \\text{ otherwise} \\end{cases}`}\n`","pinned":false,"hidden":false,"mode":"ojs"},{"id":191,"value":"viewof poissonParams = form(html`\n\n`)","pinned":false,"hidden":false,"mode":"ojs"},{"id":216,"value":"html`λ = ${poissonParams.lambda}`","pinned":false,"hidden":false,"mode":"ojs"},{"id":212,"value":"html`${pltPoisson.render('html')}`","pinned":false,"hidden":false,"mode":"ojs"},{"id":226,"value":"md` \n**Gamma Distribution**\n\n${tex.block`f(x|\\alpha,\\beta)=\\frac{\\beta^\\alpha}{\\Gamma(\\alpha)} x^{\\alpha \\,-\\, 1} e^{- \\beta x }`}\n`","pinned":false,"hidden":false,"mode":"ojs"},{"id":229,"value":"viewof gammaParams = form(html`\n\n`)","pinned":false,"hidden":false,"mode":"ojs"},{"id":232,"value":"html`\nα = ${gammaParams.alpha}
\nβ = ${gammaParams.beta}\n`","pinned":false,"hidden":false,"mode":"ojs"},{"id":240,"value":"html`${pltGamma.render('html')}`","pinned":false,"hidden":false,"mode":"ojs"},{"id":105,"value":"md` \n**Student's t-Distribution**\n\n${tex.block`f(x|\\nu) = \\frac{1} {\\sqrt{\\nu}\\,B\\left( \\tfrac{1}{2}, \\tfrac{\\nu}{2} \\right )} \\left(1+\\frac{x^2}{\\nu} \\right)^{-\\frac{\\nu+1}{2}}`}\n`","pinned":false,"hidden":false,"mode":"ojs"},{"id":110,"value":"viewof studentParams = form(html`\n\n`)","pinned":false,"hidden":false,"mode":"ojs"},{"id":118,"value":"html`${pltStudent.render('html')}`","pinned":false,"hidden":false,"mode":"ojs"},{"id":122,"value":"md` \n**F Distribution**\n\n${tex.block`f(x| d_1,d_2) = \\frac{\\sqrt{\\frac{(d_1\\,x)^{d_1}\\,\\,d_2^{d_2}} {(d_1\\,x+d_2)^{d_1+d_2}}}} {x\\,\\mathrm{B}\\!\\left(\\frac{d_1}{2},\\frac{d_2}{2}\\right)}`}\n`","pinned":false,"hidden":false,"mode":"ojs"},{"id":125,"value":"viewof fParams = form(html`\n\n`)","pinned":false,"hidden":false,"mode":"ojs"},{"id":132,"value":"html`${pltf.render('html')}`","pinned":false,"hidden":false,"mode":"ojs"},{"id":143,"value":"md` \n**${tex`\\chi^2`} Distribution**\n\n${tex.block`f(x|k) = {\\frac{1}{2^{\\frac {k}{2}}\\Gamma\\left({\\frac{k}{2}}\\right)}}\\;x^{{\\frac {k}{2}}-1}e^{-{\\frac {x}{2}}}`}\n`","pinned":false,"hidden":false,"mode":"ojs"},{"id":148,"value":"viewof x2Params = form(html`\n\n`)","pinned":false,"hidden":false,"mode":"ojs"},{"id":155,"value":"html`${pltx2.render('html')}`","pinned":false,"hidden":false,"mode":"ojs"},{"id":246,"value":"md`## Central Limit Theorem `","pinned":false,"hidden":false,"mode":"ojs"},{"id":732,"value":"$30","pinned":false,"hidden":false,"mode":"ojs"},{"id":335,"value":"{\n const bar = vl.markLine()\n .data(randUniDist)\n .encode(\n vl.x().fieldQ('X').title(null),\n vl.y().fieldQ('Y').title('PDF'),\n );\n\n const scatterplot = vl.markPoint()\n .data(randUni)\n .encode(\n vl.x().fieldQ('X').title('Index'),\n vl.y().fieldQ('Y').title('Value')\n );\n\n return vl.hconcat(bar, scatterplot)\n .config({'view':{'stroke':null},\n 'axis':{'grid':false}\n })\n .render();\n}","pinned":false,"hidden":false,"mode":"ojs"},{"id":396,"value":"{\n const dist = vl.markLine()\n .data(aq.table({\n 'X': normX,\n 'Y': normY\n }))\n .encode(\n vl.x().fieldQ('X').title(null),\n vl.y().fieldQ('Y').title('PDF'),\n )\n .config({'view':{'stroke':null},\n 'axis':{'grid':false}\n });\n\n const scatterplot = vl.markPoint()\n .data(randNormal)\n .encode(\n vl.x().fieldQ('X').title('Index'),\n vl.y().fieldQ('Y').title('Value')\n );\n\n\n return vl.hconcat(dist, scatterplot)\n .config({'view':{'stroke':null},\n 'axis':{'grid':false}\n })\n .render();\n}","pinned":false,"hidden":false,"mode":"ojs"},{"id":412,"value":"{\n const dist = vl.markLine()\n .data(aq.table({\n 'X': fX,\n 'Y': expY\n }))\n .encode(\n vl.x().fieldQ('X').title(null),\n vl.y().fieldQ('Y').title('PDF'),\n )\n .config({'view':{'stroke':null},\n 'axis':{'grid':false}\n });\n\n const scatterplot = vl.markPoint()\n .data(randExp)\n .encode(\n vl.x().fieldQ('X').title('Index'),\n vl.y().fieldQ('Y').title('Value')\n );\n\n return vl.hconcat(dist, scatterplot)\n .config({'view':{'stroke':null},\n 'axis':{'grid':false}\n })\n .render();\n}","pinned":false,"hidden":false,"mode":"ojs"},{"id":450,"value":"viewof sampleCLT = form(html`\n\n`)","pinned":false,"hidden":false,"mode":"ojs"},{"id":453,"value":"html`\nSample Size = ${sampleCLT.sampleSize} and Number of Sampling = ${sampleCLT.numberOfSampling}\n`","pinned":false,"hidden":false,"mode":"ojs"},{"id":464,"value":"vl.markBar()\n .data(sampleSet)\n .encode(\n vl.x().fieldQ('Y').bin(true),\n vl.y().count(),\n )\n .config({'view':{'stroke':null},\n 'axis':{'grid':false}\n })\n.render()","pinned":false,"hidden":false,"mode":"ojs"},{"id":252,"value":"md`## Hypothesis Testing `","pinned":false,"hidden":false,"mode":"ojs"},{"id":518,"value":"md `Tom flips one coin 30 times. 1 means head and 0 means tail.`","pinned":false,"hidden":false,"mode":"ojs"},{"id":521,"value":"Tom = [\n 1, 0, 0, 0, 0, 0, 1, 0, 0, 1,\n 0, 0, 0, 1, 0, 0, 0, 1, 0, 0,\n 0, 1, 1, 0, 0, 0, 0, 0, 0, 0\n]","pinned":false,"hidden":false,"mode":"ojs"},{"id":528,"value":"md `Jerry flips the same coin 30 times. 1 means head and 0 means tail.`","pinned":false,"hidden":false,"mode":"ojs"},{"id":531,"value":"Jerry = [\n 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,\n 0, 0, 0, 0, 0, 0, 1, 0, 0, 0,\n 0, 0, 0, 0, 0, 0, 0, 1, 0, 0\n]","pinned":false,"hidden":false,"mode":"ojs"},{"id":534,"value":"md` \nTom: Proportion of \"1\" = ${d3.format('.1%')(d3.sum(Tom) / Tom.length)}\n\nJerry: Proportion of \"1\" = ${d3.format('.1%')(d3.sum(Jerry) / Jerry.length)}\n`","pinned":false,"hidden":false,"mode":"ojs"},{"id":767,"value":"md `Let's statistically confirm if Tom has a significantly higher proportion of flipping head than Jerry.\n\nThe **null hypothesis**, often denoted ${tex`H_0`}, denotes the case where a difference is **not** statistically significant. The **alternative hypothesis**, often denoted ${tex`H_1`}, denotes the case where a difference **is** statistically significant.\n\n${tex.block`H_0: P(Tom) \\leq P(Jerry)`}\n${tex.block`H_1: P(Tom) > P(Jerry)`}\n\nThe significance level of a test (${tex`\\alpha`}), is the probability that one can reject the **null hypothesis** even though the **null hypothesis** is true. That is to say, the significance level is a measure of the strength of the evidence that must be present in sample before one will reject the null hypothesis and conclude that the effect is statistically significant.\n\n\n`","pinned":false,"hidden":false,"mode":"ojs"},{"id":781,"value":"stdlib.ttest2(Tom, Jerry, {alternative : 'greater'})","pinned":true,"hidden":false,"mode":"ojs"},{"id":798,"value":"md`## Univariate`","pinned":false,"hidden":false,"mode":"ojs"},{"id":799,"value":"sample1 = [{'ID': 3, 'ODI': 30}, {'ID': 5, 'ODI': 32},{'ID': 1, 'ODI': 40},{'ID': 6, 'ODI': 44},{'ID': 7, 'ODI': 45},{'ID': 12, 'ODI': 50},{'ID': 9, 'ODI': 52},{'ID': 11, 'ODI': 57},{'ID': 2, 'ODI': 63},{'ID': 4, 'ODI': 68},{'ID': 10, 'ODI': 82}]","pinned":true,"hidden":false,"mode":"ojs"},{"id":803,"value":"table1 = aq.from(sample1).view()","pinned":false,"hidden":false,"mode":"ojs"},{"id":14,"value":"md`## Appendix`","pinned":false,"hidden":false,"mode":"ojs"},{"id":682,"value":"md`**Library**`","pinned":false,"hidden":false,"mode":"ojs"},{"id":475,"value":"d3 = require('d3@5')","pinned":false,"hidden":false,"mode":"ojs"},{"id":19,"value":"stdlib = require( 'https://unpkg.com/@stdlib/stdlib@0.0.43' )","pinned":false,"hidden":false,"mode":"ojs"},{"id":46,"value":"import {form} from \"@mbostock/form-input\"","pinned":false,"hidden":false,"mode":"ojs"},{"id":306,"value":"import {vl} from '@vega/vega-lite-api'","pinned":false,"hidden":false,"mode":"ojs"},{"id":309,"value":"import { aq, op } from '@uwdata/arquero'","pinned":false,"hidden":false,"mode":"ojs"},{"id":443,"value":"_ = (await import(\"https://unpkg.com/underscore@1.10.1/modules/index-all.js\"))","pinned":false,"hidden":false,"mode":"ojs"},{"id":686,"value":"md`**Distribution**`","pinned":false,"hidden":false,"mode":"ojs"},{"id":86,"value":"normX = stdlib.linspace(-10, 10, 100)","pinned":false,"hidden":false,"mode":"ojs"},{"id":88,"value":"normY = normX.map(n => stdlib.base.dists.normal.pdf(n, normParams.mu, normParams.sigma))","pinned":false,"hidden":false,"mode":"ojs"},{"id":97,"value":"pltNorm = stdlib.plot({\n 'x': [ normX ], // e.g., 'x': [ x ]\n 'y': [ normY ], // e.g., 'y': [ pdf ]\n 'xLabel': 'x',\n 'yLabel': 'pdf',\n 'yMin': 0.0,\n 'width': 600\n})","pinned":false,"hidden":false,"mode":"ojs"},{"id":108,"value":"studentY = normX.map(n => stdlib.base.dists.t.pdf(n, studentParams.nu))","pinned":false,"hidden":false,"mode":"ojs"},{"id":115,"value":"pltStudent = stdlib.plot({\n 'x': [ normX ], // e.g., 'x': [ x ]\n 'y': [ studentY ], // e.g., 'y': [ pdf ]\n 'xLabel': 'x',\n 'yLabel': 'pdf',\n 'yMin': 0.0,\n 'width': 600\n})","pinned":false,"hidden":false,"mode":"ojs"},{"id":134,"value":"fX = stdlib.linspace(0, 10, 100)","pinned":true,"hidden":false,"mode":"ojs"},{"id":128,"value":"fY = fX.map(n => stdlib.base.dists.f.pdf(n, fParams.d1, fParams.d2))","pinned":false,"hidden":false,"mode":"ojs"},{"id":130,"value":"pltf = stdlib.plot({\n 'x': [ fX ], // e.g., 'x': [ x ]\n 'y': [ fY ], // e.g., 'y': [ pdf ]\n 'xLabel': 'x',\n 'yLabel': 'pdf',\n 'yMin': 0.0,\n 'width': 600\n})","pinned":false,"hidden":false,"mode":"ojs"},{"id":146,"value":"x2Y = fX.map(n => stdlib.base.dists.chisquare.pdf(n, x2Params.k))","pinned":false,"hidden":false,"mode":"ojs"},{"id":151,"value":"pltx2 = stdlib.plot({\n 'x': [ fX ], // e.g., 'x': [ x ]\n 'y': [ x2Y ], // e.g., 'y': [ pdf ]\n 'xLabel': 'x',\n 'yLabel': 'pdf',\n 'yMin': 0.0,\n 'width': 600\n})","pinned":false,"hidden":false,"mode":"ojs"},{"id":164,"value":"expY = fX.map(n => stdlib.base.dists.exponential.pdf(n, expParams.lambda))","pinned":false,"hidden":false,"mode":"ojs"},{"id":173,"value":"pltExp = stdlib.plot({\n 'x': [ fX ], // e.g., 'x': [ x ]\n 'y': [ expY ], // e.g., 'y': [ pdf ]\n 'xLabel': 'x',\n 'yLabel': 'pdf',\n 'yMin': 0.0,\n 'width': 600\n})","pinned":false,"hidden":false,"mode":"ojs"},{"id":194,"value":"poissionX = [...Array(100).keys()]","pinned":true,"hidden":false,"mode":"ojs"},{"id":202,"value":"poissionY = poissionX.map(n => stdlib.base.dists.poisson.pmf(n, poissonParams.lambda))","pinned":false,"hidden":false,"mode":"ojs"},{"id":206,"value":"pltPoisson = stdlib.plot({\n 'x': [ poissionX ], // e.g., 'x': [ x ]\n 'y': [ poissionY ], // e.g., 'y': [ pdf ]\n 'xLabel': 'x',\n 'yLabel': 'pdf',\n 'yMin': 0.0,\n 'width': 600\n})","pinned":false,"hidden":false,"mode":"ojs"},{"id":236,"value":"gammaY = poissionX.map(n => stdlib.base.dists.gamma.pdf(n, gammaParams.alpha, gammaParams.beta))","pinned":false,"hidden":false,"mode":"ojs"},{"id":238,"value":"pltGamma = stdlib.plot({\n 'x': [ poissionX ], // e.g., 'x': [ x ]\n 'y': [ gammaY ], // e.g., 'y': [ pdf ]\n 'xLabel': 'x',\n 'yLabel': 'pdf',\n 'yMin': 0.0,\n 'width': 600\n})","pinned":false,"hidden":false,"mode":"ojs"},{"id":255,"value":"poissionData = aq.table({\n 'X': poissionX,\n 'Y': gammaY\n})","pinned":false,"hidden":false,"mode":"ojs"},{"id":446,"value":"_.sample(normX, 10)","pinned":false,"hidden":false,"mode":"ojs"},{"id":690,"value":"md`**Central Limit Theorem**`","pinned":false,"hidden":false,"mode":"ojs"},{"id":693,"value":"randomNumbers = {\n let rand = [];\n for(let i=0; i<1000; i++) {\n rand.push(stdlib.base.random.randu())\n }\n return rand;\n}","pinned":false,"hidden":false,"mode":"ojs"},{"id":697,"value":"randUni = aq.table({\n 'X':[...Array(1000).keys()] ,\n 'Y':randomNumbers\n})","pinned":false,"hidden":false,"mode":"ojs"},{"id":701,"value":"randUniDist = aq.table({\n 'X': randomNumbers,\n 'Y': randomNumbers.map(n => stdlib.base.dists.uniform.pdf(n, 0,2))\n})","pinned":false,"hidden":false,"mode":"ojs"},{"id":705,"value":"randomNumbersNorm = {\n let rand = [];\n for(let i=0; i<1000; i++) {\n rand.push(stdlib.base.random.normal(normParams.mu, normParams.sigma))\n }\n return rand;\n}","pinned":false,"hidden":false,"mode":"ojs"},{"id":708,"value":"randNormal = aq.table({\n 'X':[...Array(1000).keys()] ,\n 'Y':randomNumbersNorm\n})","pinned":false,"hidden":false,"mode":"ojs"},{"id":713,"value":"randomNumbersExp = {\n let rand = [];\n for(let i=0; i<1000; i++) {\n rand.push(stdlib.base.random.exponential(expParams.lambda))\n }\n return rand;\n}","pinned":false,"hidden":false,"mode":"ojs"},{"id":717,"value":"randExp = aq.table({\n 'X':[...Array(1000).keys()] ,\n 'Y':randomNumbersExp\n})","pinned":false,"hidden":false,"mode":"ojs"},{"id":721,"value":"sample = {\n let s = [];\n for (let i=0; i n.Y), sampleCLT.sampleSize)))\n }\n return s;\n}","pinned":false,"hidden":false,"mode":"ojs"},{"id":725,"value":"sampleSet = aq.table({\n 'X': [...Array(sample.length).keys()],\n 'Y': sample\n})","pinned":false,"hidden":false,"mode":"ojs"},{"id":729,"value":"datasets = { return {randUni: randUni, randNormal: randNormal, randExp, randExp}}","pinned":true,"hidden":false,"mode":"ojs"}],"files":"$W31","stdlib":"1","titling":"auto"}