2a:null 2b:["$","title",null,{"children":"Continuous random variables | Pierre PC | Observable"}] 30:T4cc,drawExperiment = function (context, sizex, sizey, hline, left, right, size, data, options) { const opts = (options === undefined)?{}:options; const defaultOpts = [['color','black'],['visibleTicks',true]] defaultOpts.forEach(function (opt) { opts[opt[0]] = (opt[0] in opts)?opts[opt[0]]:opt[1] }) const lwidth = right - left; const radius = size/2; const textSize = Math.floor(4*size/3); const bg_elements = [ [[left,hline],[right,hline]], [[left,hline+size],[left,hline-size]], [[right,hline+size],[right,hline-size]] ]; bg_elements.forEach(function (l) { context.beginPath(); context.moveTo(...l[0]); context.lineTo(...l[1]); context.stroke() }); context.fillStyle = opts.color; context.textAlign = "center"; context.textBaseline = "middle"; context.font = `${textSize}px Arial`; context.fillText("0",left/2,hline); context.fillText("1",(sizex+right)/2,hline); context.globalAlpha = 0.5; if (opts.visibleTicks) { data.forEach(function (x) { context.beginPath(); context.arc(left + lwidth*x, hline, radius, 0, 2 * Math.PI); context.fill() }); } context.globalAlpha = 1; }31:T893,drawHistogram = function (context, sizex, sizey, top, bottom, left, right, size, data, k, options) { const opts = (options === undefined)?{}:options; const defaultOpts = [['color','black'],['visibleTicks',false],['histColor','black'],['yLabel','left']] defaultOpts.forEach(function (opt) { opts[opt[0]] = (opt[0] in opts)?opts[opt[0]]:opt[1] }) const lWidth = right - left; const vHeight = bottom - top; const radius = size/2; const textSize = Math.floor(4*size/3); const labelPosition = (opts.yLabel === 'left')?left:((opts.yLabel === 'right')?right:sizex); const graduations = Array.from({length: k+1}, function (v,i) {return [[left+i*lWidth/k,bottom+size], [left+i*lWidth/k,bottom-size]]}); graduations.forEach(function (l) { context.beginPath(); context.moveTo(...l[0]); context.lineTo(...l[1]); context.stroke() }); const heights = data.reduce(function (total,current) { var next = total; next[Math.floor(k*current)]++; return next }, Array.from({length: k}, () => 0)); const max = Math.max(...heights); const unitHeight = vHeight/max; const pGraduation = euroFloor((max/data.length)/3); const nGraduation = pGraduation*data.length; const gHeight = nGraduation*unitHeight; context.fillStyle = opts.histColor; context.globalAlpha = 0.6; heights.forEach(function (h,i) { context.fillRect(left+i*lWidth/k, bottom, lWidth/k, -h*vHeight/max) }); context.globalAlpha = 1; context.textAlign = (opts.yLabel === 'right')?'right':'left'; context.textBaseline = 'bottom'; context.font = `${textSize}px Arial`; for (var i=1; gHeight*i<=vHeight; i++) { context.beginPath(); context.moveTo(left,bottom-i*gHeight); context.lineTo(right,bottom-i*gHeight); context.stroke() context.fillText(`${100*i*pGraduation}%`,labelPosition,bottom-i*gHeight); } drawExperiment(context,sizex,sizey,bottom,left,right,size,data,opts); }32:[] 2e:{"id":"482414ac3207b588","slug":"continuous-random-variables","title":"Continuous random variables","thumbnail":"4cf966f8e0e30bea99ea979df322328ad209718ecb0d09ced455485930c32901","access_level":"public","create_time":"$D2020-03-18T22:08:28.137Z","update_time":"$D2020-03-23T16:30:31.364Z","view_time":null,"first_public_time":"$D2020-03-23T16:30:41.078Z","fork":false,"stars":0,"starred":false,"owner":{"type":"team","id":"e63037398666117f","login":"pierrepc","name":"Pierre PC","avatar_url":"https://avatars-next.observableusercontent.com/22250a38885985472d38da2a64469ebd2a0f1b7ba0e6869a5ccb10ce8ca7aa66","tier":"starter_2024"},"version":732,"publish_version":732,"role":null,"authored":false,"authors":[{"type":"individual","id":"e1fb4a1ab7bf50c7","login":"pierrepc","name":"Pierre PC","avatar_url":"https://avatars-next.observableusercontent.com/22250a38885985472d38da2a64469ebd2a0f1b7ba0e6869a5ccb10ce8ca7aa66","tier":null}],"members":[{"type":"individual","id":"e1fb4a1ab7bf50c7","login":"pierrepc","name":"Pierre PC","avatar_url":"https://avatars-next.observableusercontent.com/22250a38885985472d38da2a64469ebd2a0f1b7ba0e6869a5ccb10ce8ca7aa66","tier":null,"role":"editor"}],"comments":[],"custom_thumbnail":false,"trashed":false,"subscribed":null,"forks":0,"collection":{"id":"ac21ca0b7e8ed20d","slug":"probability-course","title":"Probability course","owner":{"id":"e63037398666117f","login":"pierrepc","name":"Pierre PC"}},"type":"notebook","body":{"title":"Continuous random variables","cells":[{"id":0,"value":"md`# Continuous random variables`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":432,"value":"md`A continuous variable can take any value in an interval. But maybe some values are more likely than others.\n\nBelow are example when the random variables seem evenly distributed, or accumulate on the left or the right. How can we give rigorous meaning to these notions?`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":7,"value":"N = { const elmt = html`

Number of experiments: 10
\n
\n

`;\n elmt.range = elmt.getElementsByTagName('input')[0];\n elmt.button = elmt.getElementsByTagName('input')[1];\n elmt.span = elmt.getElementsByTagName('span')[0];\n elmt.button.onclick = function () {\n master.updateAll(elmt.value,'hard')\n };\n elmt.range.oninput = function () {\n elmt.value = elmt.range.value;\n elmt.span.innerHTML = elmt.value;\n master.updateAll(elmt.value,'soft')\n };\n elmt.range.oninput();\n return elmt\n }","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":515,"value":"md`## 1. A few samples of distribution that look different`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":436,"value":"md`We simulate ${N.value} independent random variables. It seems that most values are on the left half for the “Heavy left” experiment below.\n\nTry reloading a few times to test this theory.`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":164,"value":"reloadButton('Uniform random variable', uniform, histUniform)","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":30,"value":"uniform = experiment((x) => x, 'black')","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":185,"value":"reloadButton('Heavy left random variable', heavyleft, histHeavyleft)","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":187,"value":"heavyleft = experiment((x) => x*x, 'red')","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":217,"value":"reloadButton('Heavy right random variable', heavyright, histHeavyright)","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":220,"value":"heavyright = experiment((x) => 1-x*x,'blue')","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":610,"value":"md`## 2. How different are these distributions?`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":445,"value":"md`Let us actually count the number of times the random variables fall into a given interval. When the number of experiments is large, the biais becomes undeniable.`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":391,"value":"reloadButton('Uniform random variable', uniform, histUniform)","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":386,"value":"histUniform = histogram(uniform, {histColor: 'black', visibleTicks: true})","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":406,"value":"reloadButton('Heavy left random variable', heavyleft, histHeavyleft)","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":402,"value":"histHeavyleft = histogram(heavyleft, {histColor: 'red', visibleTicks: true, yLabel: 'right'})","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":420,"value":"reloadButton('Heavy right random variable', heavyright, histHeavyright)","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":415,"value":"histHeavyright = histogram(heavyright, {histColor: 'blue', visibleTicks: true})","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":395,"value":"reloadButton = function (title,target,histTarget) {\n const elmt = html`

${title}

\n `;\n elmt.getElementsByTagName('input')[0].onclick = function () {\n target.update(N.value,'hard');\n histTarget.update(target.Nvisible,10)\n };\n return elmt\n }","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":197,"value":"experiment = function (f, color) {\n const canvas = document.createElement('canvas');\n const context = canvas.getContext('2d');\n canvas.width = width;\n const h = 100;\n canvas.height = h;\n \n canvas.update = function (n,option) {\n if (option === 'hard') {\n canvas.data = Array.from({length: Nmax}, function () {return f(Math.random())})\n }\n canvas.Nvisible = n;\n context.clearRect(0,0,width,h);\n drawExperiment(context, width, h, h/2, width/30, 29*width/30, h/6, canvas.data.slice(0,canvas.Nvisible),{color: color});\n };\n \n canvas.update(10,'hard');\n \n return canvas\n }","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":380,"value":"histogram = function (exp, options) {\n const canvas = document.createElement('canvas');\n const context = canvas.getContext('2d');\n canvas.width = width;\n canvas.height = height;\n const sizex = width;\n const sizey = height;\n const top = 50;\n const bottom = height - top;\n const left = width/30;\n const right = 29*width/30;\n const size = top/3;\n \n canvas.experiment = exp;\n \n canvas.update = function (n,k) {\n context.clearRect(0,0,sizex,sizey);\n \n const slicedData = canvas.experiment.data.slice(0,n);\n \n drawHistogram(context, sizex, sizey, top, bottom, left, right, size, slicedData, k, options)\n };\n \n canvas.update(exp.Nvisible,10);\n \n return canvas\n }","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":20,"value":"$30","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":357,"value":"$31","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":623,"value":"euroFloor = function (x) {\n var y = 10**Math.floor(Math.log10(x));\n y = (x>=5*y)?5*y:y;\n y = (x>=2*y)?2*y:y;\n return y\n }","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":301,"value":"master = ({\n updateAll: function (n,option) {\n uniform.update(n,option);\n heavyleft.update(n,option);\n heavyright.update(n,option);\n \n histUniform.update(n,10);\n histHeavyleft.update(n,10);\n histHeavyright.update(n,10);\n }\n })","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":118,"value":"Nmax = 100","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":243,"value":"height = 600","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"}],"files":"$W32","stdlib":"1","titling":"auto"}} 2f:"$2e:body"