2b:null 2c:["$","title",null,{"children":"Expected value as simple sum | Sander Evers | Observable"}] 31:[] 2f:{"id":"4f829e6802a86388","slug":"expected-value-as-simple-sum","title":"Expected value as simple sum","thumbnail":"a289dd7d70836ff25074fa6ca88b200c15d61e9263da1cd30c5a7b224f40a59f","access_level":"public","create_time":"$D2024-10-02T05:29:29.467Z","update_time":"$D2025-02-19T09:19:39.279Z","view_time":null,"first_public_time":"$D2024-10-03T06:00:05.009Z","fork":false,"stars":0,"starred":false,"owner":{"type":"team","id":"f8968868603f3a63","login":"sanderevers","name":"Sander Evers","avatar_url":"https://avatars-next.observableusercontent.com/368f20f13b863df596105f72d5a25388cc1a7808dc9fa0ee74858280b5847b35","tier":"starter_2024"},"version":442,"publish_version":null,"role":null,"authored":false,"authors":[{"type":"individual","id":"f9fc95e35699f5b9","login":"sanderevers","name":"Sander Evers","avatar_url":"https://avatars-next.observableusercontent.com/368f20f13b863df596105f72d5a25388cc1a7808dc9fa0ee74858280b5847b35","tier":null}],"members":[{"type":"individual","id":"f9fc95e35699f5b9","login":"sanderevers","name":"Sander Evers","avatar_url":"https://avatars-next.observableusercontent.com/368f20f13b863df596105f72d5a25388cc1a7808dc9fa0ee74858280b5847b35","tier":null,"role":"editor"},{"type":"team","id":"f8968868603f3a63","login":"sanderevers","name":"Sander Evers","avatar_url":"https://avatars-next.observableusercontent.com/368f20f13b863df596105f72d5a25388cc1a7808dc9fa0ee74858280b5847b35","tier":"starter_2024","role":"viewer"}],"comments":[],"custom_thumbnail":true,"trashed":false,"subscribed":null,"forks":0,"collection":null,"type":"notebook","body":{"title":"Expected value as simple sum","cells":[{"id":0,"value":"# Expected value as simple sum","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined"},{"id":264,"value":"Did you know...\n\n> if you have a discrete distribution over ${tex`X=0,1,2,3,\\ldots,n`}\n\nyou can calculate the expected value by simply adding cumulative probabilities ","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined"},{"id":280,"value":"\\mathbb{E}(X) = P(X \\geq 1) + P(X \\geq 2) + P(X \\geq 3) + \\ldots + P(X \\geq n)","pinned":false,"hidden":false,"mode":"tex","author":"$undefined","summary":"$undefined"},{"id":296,"value":"? (I did not.) Let's take this example:","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined"},{"id":243,"value":"viewof table = Inputs.table(vals.map(v => ({x:v,'P(X=x)':dist[v],'P(X≥x)':rcdist[v]})),{select:false,layout:'auto',maxWidth:'300px'})","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":310,"value":" \n\n### Here's a visual proof","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined"},{"id":48,"value":"","pinned":false,"hidden":false,"mode":"html","author":"$undefined","summary":"$undefined"},{"id":312,"value":"These two areas are equal:","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined"},{"id":117,"value":" mutable choice=0}>\n${rcdist.flatMap((p,i) => [' + ',html`${p}`]).slice(3)} = ${rc_avg}","pinned":false,"hidden":false,"mode":"html","author":"$undefined","summary":"$undefined"},{"id":126,"value":" mutable choice=1}> (\n${dist.flatMap((p,i) => [i==1?' +) ':' + ',html`${p}·${i}`]).slice(1)} = ${avg}","pinned":false,"hidden":false,"mode":"html","author":"$undefined","summary":"$undefined"},{"id":409,"value":" \n\n### And a proofy proof","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined"},{"id":341,"value":"tex`\\phantom{=}\\mathbb{E}(X)\\\\\n=\\sum_{0\\leq x \\leq n} P(X=x)\\cdot x\\\\\n=\\sum_{0\\leq x \\leq n} P(X \\geq x)\\cdot x - P(X \\geq x{+}1)\\cdot x\n`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":392,"value":"tex`=P(X \\geq 0) \\cdot 0 + P(X \\geq 1)\\cdot 1 + P(X \\geq 2)\\cdot 2 + \\ldots + P(X \\geq n) \\cdot n\\\\\n\\phantom{=P(X \\geq 0)\\cdot 0}- P(X \\geq 1)\\cdot 0 - P(X \\geq 2)\\cdot 1 - \\ldots - P(X \\geq n) \\cdot (n{-}1) - \\underbrace{P(X \\geq n{+}1)}_{0} \\cdot n\\\\\n=\\phantom{P(X \\geq 0)\\cdot 0 -} P(X \\geq 1)\\phantom{\\cdot 1} + P(X \\geq 2)\\phantom{\\cdot 2} + \\ldots + P(X \\geq n)\n`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":435,"value":" \n\n# ❦","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined"},{"id":414,"value":" \n\n \n\n \n\n \n\n \n\n \n\n \n","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined"},{"id":430,"value":"

Here be dragons

","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined"},{"id":5,"value":"vals = d3.range(5)","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":7,"value":"dist = [0.2,0.3,.15,.25,.1]","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":13,"value":"avg = d3.sum(d3.zip(vals,dist).map(([a,b])=>a*b))","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":9,"value":"rcdist = d3.reverse(d3.cumsum(d3.reverse(dist)))","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":147,"value":"mutable choice=0","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":50,"value":"makeRects = (val) => [...rcdist,0].slice(val+1).map((cprob,i) => \n svg``)","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":182,"value":"makeText = (x,y,t,s) => svg`\n${t}`","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":92,"value":"rwidth = 1/vals.length","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":16,"value":"rc_avg = d3.sum(rcdist.slice(1))","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":19,"value":"cdist = d3.cumsum(dist)","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":21,"value":"c_avg = 4 - d3.sum(cdist.slice(0,-1))","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":31,"value":"color = d3.scaleOrdinal(d3.schemeCategory10)","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":109,"value":"rcolor = (v,i) => v==0?'#eeeeee': color(choice?i:v)","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":156,"value":"tcolor = (nr,i) => (choice==nr && i>0) ? color(i) : \"#000000\"","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"}],"files":"$W31","stdlib":"1","titling":"auto"}} 30:"$2f:body"