2b:null 2c:["$","title",null,{"children":"Bernoulli distribution | Mattias Villani | Observable"}] 31:[] 2f:{"id":"41d44cc710c99378","slug":"bernoulli-distribution","title":"Bernoulli distribution","thumbnail":"7792c9e78656ef0605845030c5aacc4988de2dda597dba2d0005e7d38f4c6872","access_level":"public","create_time":"$D2023-02-10T10:57:10.753Z","update_time":"$D2023-09-04T09:17:45.153Z","view_time":null,"first_public_time":"$D2023-02-10T10:57:10.753Z","fork":{"id":"8e97614a2bda7e05","slug":"binomial-distribution","title":"Binomial distribution","owner":{"id":"7592a7ab121f74a0","login":"mattiasvillani","name":"Mattias Villani"}},"stars":1,"starred":false,"owner":{"type":"team","id":"7592a7ab121f74a0","login":"mattiasvillani","name":"Mattias Villani","avatar_url":"https://avatars-next.observableusercontent.com/a440aaf39a3abefb86cbcc758fd7a5ab42555ae68d6825b52ec006f385a94d1d","tier":"pro_2024"},"version":904,"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"}],"comments":[],"custom_thumbnail":false,"trashed":false,"subscribed":null,"forks":2,"collection":{"id":"9fdb7c9ba7be6ba8","slug":"bayesian-learning","title":"Bayesian Learning","owner":{"id":"7592a7ab121f74a0","login":"mattiasvillani","name":"Mattias Villani"}},"type":"notebook","body":{"title":"Bernoulli distribution","cells":[{"id":38,"value":"# Bernoulli distribution","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined"},{"id":824,"value":"textdist = md`

${tex`X\\sim \\operatorname{Bern}(p)`}, for ${tex`X \\in \\{0,1\\}`}`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":496,"value":"### Probability function and moments","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined"},{"id":118,"value":"viewof p = Inputs.range([0, 1], {value: 0.5, step: 0.01, label: tex`p`})\n","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":895,"value":"momentstext = tex`\\text{If } X \\sim \\operatorname{Bern}(${p}) \\text{ then } `","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":898,"value":"moments = tex`\n\\begin{aligned}\n&E( X) =p = ${p} \\\\[1em]\n&Var( X) = p(1-p) = ${(p*(1-p)).toPrecision(3)} \\\\[1em]\n\\end{aligned}\n`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":305,"value":"plt_pdf = Plot.plot({\n x: {\n label: \"x\",\n axis: true,\n domain: [-0.5,1.5],\n ticks: [0,1],\n tickFormat: \".0\"\n },\n y: {\n label: \"P(X=x)\",\n domain: [0, 1.05]\n },\n marks: [\n Plot.ruleY([0]),\n Plot.ruleX([-0.5]),\n Plot.ruleX([{x: 0, y: 1-p},{x: 1, y: p}], {x: \"x\", y: \"y\", stroke: \"steelblue\", strokeWidth: 2}), \n Plot.dot([{x: 0, y: 1-p},{x: 1, y: p}], {x: \"x\", y: \"y\", fill: \"steelblue\", r: 5}),\n Plot.text([{x: 0, y: 1-p},{x: 1, y: p}], {x: \"x\", y: \"y\", text: \"y\", dy: -15})\n ]\n })","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":109,"value":"
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\n\n---\n## 🙈 Appendix 🙈\n\nJust some implementation details below here that you probably don't want to read.","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined"},{"id":229,"value":"#### Style","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined"},{"id":202,"value":"","pinned":false,"hidden":false,"mode":"html","author":"$undefined","summary":"$undefined"},{"id":58,"value":"viewof md_colour = Inputs.color({label: \"Highlight color cells\", value: \"#F5F5DC\"})","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"}],"files":"$W31","stdlib":"1","titling":"auto"}} 30:"$2f:body"