// Numbas version: finer_feedback_settings
{"name": "20122013 CBA1_1", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"parts": [{"scripts": {}, "gaps": [{"showPrecisionHint": false, "scripts": {}, "allowFractions": false, "type": "numberentry", "correctAnswerFraction": false, "minValue": "ans1-tol4", "showCorrectAnswer": true, "marks": 1, "maxValue": "ans1+tol4"}, {"showPrecisionHint": false, "scripts": {}, "allowFractions": false, "type": "numberentry", "correctAnswerFraction": false, "minValue": "ans2-tol3", "showCorrectAnswer": true, "marks": 1, "maxValue": "ans2+tol3"}], "type": "gapfill", "showCorrectAnswer": true, "prompt": "<p>For $\\mu=\\var{m}$ calculate:</p>\n        <p>(i) $L(\\mu,\\underline{x})=\\;$? [[0]] (enter your answer to 4 decimal places).</p>\n        <p>(ii) $l(\\mu,\\underline{x})=\\;$? [[1]] (enter your answer to 3 decimal places).</p>", "marks": 0}, {"scripts": {}, "gaps": [{"showPrecisionHint": false, "scripts": {}, "allowFractions": false, "type": "numberentry", "correctAnswerFraction": false, "minValue": "ans3-tol3", "showCorrectAnswer": true, "marks": 1, "maxValue": "ans3+tol3"}], "type": "gapfill", "showCorrectAnswer": true, "prompt": "<p>The maximum likelihood estimator $\\hat{\\mu}$ for $\\mu$ &nbsp;is: [[0]]</p>\n      <p>&nbsp;(enter your answer as a decimal to 3 decimal places)</p>", "marks": 0}], "variables": {"tol3": {"group": "Ungrouped variables", "templateType": "anything", "definition": "0.001", "name": "tol3", "description": ""}, "x2": {"group": "Ungrouped variables", "templateType": "anything", "definition": "random(0..5)", "name": "x2", "description": ""}, "x1": {"group": "Ungrouped variables", "templateType": "anything", "definition": "random(0..5 except x0)", "name": "x1", "description": ""}, "v": {"group": "Ungrouped variables", "templateType": "anything", "definition": "rowvector(x0,x1,x2)", "name": "v", "description": ""}, "ans1": {"group": "Ungrouped variables", "templateType": "anything", "definition": "precround(p[0]*p[1]*p[2],4)", "name": "ans1", "description": ""}, "x0": {"group": "Ungrouped variables", "templateType": "anything", "definition": "random(0..5)", "name": "x0", "description": ""}, "ans3": {"group": "Ungrouped variables", "templateType": "anything", "definition": "precround((x0+x1+x2)/3,3)", "name": "ans3", "description": ""}, "tol4": {"group": "Ungrouped variables", "templateType": "anything", "definition": "0.0001", "name": "tol4", "description": ""}, "p": {"group": "Ungrouped variables", "templateType": "anything", "definition": "[e^(-m)*m^x0/fact(x0),e^(-m)*m^x1/fact(x1),e^(-m)*m^x2/fact(x2)]", "name": "p", "description": ""}, "ans2": {"group": "Ungrouped variables", "templateType": "anything", "definition": "precround(ln(p[0]*p[1]*p[2]),3)", "name": "ans2", "description": ""}, "m": {"group": "Ungrouped variables", "templateType": "anything", "definition": "random(1..3#0.5)", "name": "m", "description": ""}}, "ungrouped_variables": ["tol3", "ans1", "ans2", "ans3", "m", "tol4", "p", "v", "x2", "x0", "x1"], "question_groups": [{"pickingStrategy": "all-ordered", "questions": [], "name": "", "pickQuestions": 0}], "name": "20122013 CBA1_1", "functions": {}, "variable_groups": [], "variablesTest": {"condition": "", "maxRuns": 100}, "statement": "<p>Suppose $\\underline{x}=\\var{v}$&nbsp;is a vector of observations from a $\\operatorname{Poisson}(\\mu)$ distribution.</p>", "tags": ["MAS2302", "MLE", "Poisson", "Poisson parameter", "checked2015", "distributions", "likelihood", "log likelihood", "maximum likelihood estimator", "mle", "statistical distributions", "statistics"], "rulesets": {}, "preamble": {"css": "", "js": ""}, "type": "question", "metadata": {"notes": "<p><strong>26/01/2013:</strong></p>\n        <p>First draft created.</p>", "licence": "Creative Commons Attribution 4.0 International", "description": "<p>Given 3 observations from a $\\operatorname{Poisson}(\\mu)$ distribution find the likelihood, the log likelihood and the MLE for $\\mu$.</p>"}, "advice": "<p>a) If there are 3 observations from $\\operatorname{Poisson}(\\mu),\\;(x_1,x_2,x_3)$ then:</p>\n    <p>(i) \\[\\operatorname{L}(\\mu|\\underline{x})=\\frac{e^{-\\mu}\\mu^{x_1}}{x_1!}\\times\\frac{e^{-\\mu}\\mu^{x_2}}{x_2!}\\times \\frac{e^{-\\mu}\\mu^{x_3}}{x_3!}\\]</p>\n    <p>For this calculation $\\mu=\\var{m}$ and $(x_1,x_2,x_3)=\\var{v}$ and we obtain:</p>\n    <p>&nbsp;\\[\\operatorname{L}(\\mu|\\underline{x})=\\var{p[0]}\\times\\var{p[1]}\\times \\var{p[2]}=\\var{ans1}\\] to 4 decimal places.</p>\n    <p>(ii) &nbsp;\\[\\operatorname{l}(\\mu|\\underline{x})=\\ln(\\operatorname{L}(\\mu|\\underline{x}))=\\ln(\\var{p[0]}\\times\\var{p[1]}\\times \\var{p[2]})=\\var{ans2}\\]&nbsp;</p>\n    <p>to 3 decimal places.</p>\n    <p>b) The MLE is the mean of the observations i.e. \\[\\frac{\\var{x0}+\\var{x1}+\\var{x2}}{3} = \\var{ans3}\\] to 3 decimal places.</p>\n    <p>&nbsp;</p>", "showQuestionGroupNames": false, "contributors": [{"name": "Newcastle University Mathematics and Statistics", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/697/"}]}]}], "contributors": [{"name": "Newcastle University Mathematics and Statistics", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/697/"}]}