2b:null 2c:["$","title",null,{"children":"Sampling distribution and likelihood function - Normal model | Mattias Villani | Observable"}] 32:T428,Assume random sample of size *n* from a normal population with mean ${tex`\mu`} and variance ${tex`\sigma^2`}. Assume that the variance is known. ${tex.block`X_1,X_2,\ldots,X_n \overset{\mathrm{iid}}{\sim} \operatorname{N}(\mu,\sigma^2)`} The sampling distribution for the sample mean ${tex`p(\bar X \vert \mu)`} can be looked at from two viewpoints: - **For a given ${tex`\mu`}** we obtain the sampling distribution of the sample mean ${tex.block`\bar X \sim \operatorname{N}(\mu,\sigma^2/n)`} - **For a given observed sample mean ${tex`\bar X = \bar x`}** the likelihood function is the probability of the observed sample mean ${tex`\operatorname{N}(\bar x \vert \mu,\sigma^2/n)`} as function of ${tex`\mu.`} Here ${tex`\operatorname{N}(\bar x \vert \mu,\sigma^2/n)`} denotes the probability density function of the ${tex`\operatorname{N}(\mu,\sigma^2/n)`} distribution evaluated at ${tex`\bar x`}. The widget below allows you to inspect the sampling distribution and the likelihood function for different ${tex`\mu`} and observed ${tex`\bar X = \bar x`}.33:[] 2f:{"id":"9cc83e4830769ba8","slug":"sampling-distribution-and-likelihood-function-normal","title":"Sampling distribution and likelihood function - Normal model","thumbnail":"ec2bb737a8d34153a875e252327bc0ab18fef1c4e96e5cac55bf183f25b4e113","access_level":"public","create_time":"$D2024-07-27T19:46:21.828Z","update_time":"$D2024-07-28T12:26:22.361Z","view_time":null,"first_public_time":"$D2024-07-27T21:15:14.836Z","fork":{"id":"38525a45d36d7a54","slug":"ident-mean-ar-1-process","title":"Identification of the mean in an AR(1) process","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":541,"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":1,"collection":{"id":"3247547e8812c274","slug":"statistics","title":"Basic Statistics","owner":{"id":"7592a7ab121f74a0","login":"mattiasvillani","name":"Mattias Villani"}},"type":"notebook","body":{"title":"Sampling distribution and likelihood function - Normal model","cells":[{"id":0,"value":"# Sampling distribution and likelihood function - Normal model","pinned":false,"hidden":false,"mode":"md"},{"id":179,"value":"$32","pinned":false,"hidden":false,"mode":"md"},{"id":347,"value":"viewof settings = Inputs.form({\n mu: Inputs.range([-5, 5], {value: 0, label: tex`\\mu`, step: 0.1}),\n sigma: Inputs.range([0.01, 3], {value: 1, label: tex`\\sigma`, step: 0.01}),\n n: Inputs.range([1, 1000], {value: 10, label: tex`n`, step: 1}),\n xbar: Inputs.range([-5, 5], {value: 1, label: tex`\\bar x`, step: 0.1}),\n})","pinned":false,"hidden":false,"mode":"ojs"},{"id":125,"value":"jointheat = Plot.plot({\n title: \"Sampling distribution\",\n width: 500, // Set the width\n height: 500, // Set the height equal to the width for a square aspect\n x: {label: \"μ\", domain: [-5,5]},\n y: {label: \"x̄\", domain: [-5,5]},\n color: {\n legend: true,\n scheme: \"Blues\",\n //type: \"identity\",\n label: \"density\",\n //symmetric: false,\n domain: [0,sampDistMax]\n //transform: d => Math.exp(d - maxloglikIntercept)\n },\n marks: [\n Plot.contour(sampDist, {x: \"mu\", y: \"xbar\", fill: \"prob\", blur: 4, thresholds: 100}),\n //Plot.dot([{mu: mu, xbar: xbar}], {x: \"mu\", y: \"xbar\", fill: \"orange\", r: 4}),\n Plot.tip(sampDist, Plot.pointerX({x: \"mu\", y: \"xbar\", stroke: \"prob\"})),\n Plot.ruleY([xbar],{stroke: \"orange\", strokeWidth: 3}),\n Plot.ruleX([mu],{stroke: \"#8B0000\", strokeWidth: 3})\n ]\n})","pinned":false,"hidden":false,"mode":"ojs"},{"id":466,"value":"bothplots = html`\n \n \n \n

\n ${plt_samp}\n

\n ${plt_like}\n

\n \n`","pinned":false,"hidden":false,"mode":"ojs"},{"id":5,"value":"
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\n\n---\n#### 🙈 Appendix 🙈\n\nHere is just some code and details that you probably don't want to see.","pinned":false,"hidden":false,"mode":"md"},{"id":535,"value":"#### Style","pinned":false,"hidden":false,"mode":"md"},{"id":537,"value":"html`