// Numbas version: finer_feedback_settings
{"name": "Coursework 2", "metadata": {"description": "", "licence": "None specified"}, "duration": 0, "percentPass": 0, "showQuestionGroupNames": false, "shuffleQuestionGroups": false, "showstudentname": true, "question_groups": [{"name": "Group", "pickingStrategy": "all-ordered", "pickQuestions": 1, "questionNames": ["", "", "", "", "", ""], "variable_overrides": [[], [], [], [], [], []], "questions": [{"name": "InterimTest - Question 2 as 2021", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Tatiana Tyukina", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/341/"}], "tags": [], "metadata": {"description": "", "licence": "None specified"}, "statement": "<p>Let $Y_1, Y_2, . . . , Y_n$ denote a random sample from the uniform distribution on the interval $[0, t]$ with the probability density function of the form:</p>\n<p>\\[f_Y(y)=\\frac{1}{t}, \\quad 0\\leqslant y \\leqslant t.\\]</p>\n<p></p>", "advice": "", "rulesets": {}, "builtin_constants": {"e": true, "pi,\u03c0": true, "i": true}, "constants": [], "variables": {"x1": {"name": "x1", "group": "Ungrouped variables", "definition": "random(0 .. 5#1)", "description": "", "templateType": "randrange", "can_override": false}, "x3": {"name": "x3", "group": "Ungrouped variables", "definition": "random(1..5)", "description": "", "templateType": "anything", "can_override": false}, "x2": {"name": "x2", "group": "Ungrouped variables", "definition": "random(0..n-1)", "description": "", "templateType": "anything", "can_override": false}, "n": {"name": "n", "group": "Ungrouped variables", "definition": "3", "description": "", "templateType": "number", "can_override": false}, "sample": {"name": "sample", "group": "Ungrouped variables", "definition": "\"X1, X2, X3\"", "description": "", "templateType": "anything", "can_override": false}, "mu1": {"name": "mu1", "group": "Ungrouped variables", "definition": "expression(\"t/2\")", "description": "", "templateType": "anything", "can_override": false}, "mu2": {"name": "mu2", "group": "Ungrouped variables", "definition": "expression(\"t^2/3\")", "description": "", "templateType": "anything", "can_override": false}, "vary": {"name": "vary", "group": "Ungrouped variables", "definition": "expression(\"t^2/12\")", "description": "", "templateType": "anything", "can_override": false}, "t_est": {"name": "t_est", "group": "Ungrouped variables", "definition": "siground(sum(vector([x1,x2,x3]))/3*2,4)", "description": "", "templateType": "anything", "can_override": false}}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": ["x1", "x2", "x3", "n", "sample", "mu1", "mu2", "vary", "t_est"], "variable_groups": [], "functions": {}, "preamble": {"js": "", "css": ""}, "parts": [{"type": "gapfill", "useCustomName": true, "customName": "Part 1.", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<p>&nbsp;Using the first two population moments, find the mean and the variance of $Y$.&nbsp;</p>\n<p><em>Write your answer as an experssion in terms of <strong>t</strong></em></p>\n<p>$\\mu_1=E(Y)=$[[0]]</p>\n<p>$\\mu_2=$[[1]]</p>\n<p>Var$(Y)=$[[2]]</p>\n<p></p>", "gaps": [{"type": "jme", "useCustomName": false, "customName": "", "marks": "2", "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "answer": "{mu1}", "showPreview": true, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": false, "singleLetterVariables": false, "allowUnknownFunctions": true, "implicitFunctionComposition": false, "caseSensitive": false, "valuegenerators": []}, {"type": "jme", "useCustomName": false, "customName": "", "marks": "2", "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "answer": "{mu2}", "showPreview": true, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": false, "singleLetterVariables": false, "allowUnknownFunctions": true, "implicitFunctionComposition": false, "caseSensitive": false, "valuegenerators": []}, {"type": "jme", "useCustomName": false, "customName": "", "marks": "2", "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "answer": "{varY}", "showPreview": true, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": false, "singleLetterVariables": false, "allowUnknownFunctions": true, "implicitFunctionComposition": false, "caseSensitive": false, "valuegenerators": []}], "sortAnswers": false}, {"type": "gapfill", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<p>Write the first sample moment $m_1$ as a function of the sample mean, use $Ybar$ to denote $\\bar{Y}$:</p>\n<p>$m_1=$[[0]]</p>\n<p>Write the estimator for t in terms of $\\bar{Y}$:</p>\n<p>$\\hat{t}=$[[1]]</p>", "gaps": [{"type": "jme", "useCustomName": false, "customName": "", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "answer": "Ybar", "showPreview": true, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": false, "singleLetterVariables": false, "allowUnknownFunctions": true, "implicitFunctionComposition": false, "caseSensitive": false, "valuegenerators": [{"name": "ybar", "value": ""}]}, {"type": "jme", "useCustomName": false, "customName": "", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "answer": "2Ybar", "showPreview": true, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": false, "singleLetterVariables": false, "allowUnknownFunctions": true, "implicitFunctionComposition": false, "caseSensitive": false, "valuegenerators": [{"name": "ybar", "value": ""}]}], "sortAnswers": false}, {"type": "gapfill", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<p>Find the estimate of $t$ using the method of moments if&nbsp; the following numbers were observed:</p>\n<p>\\[y_1=\\var{x1},\\ y_2=\\var{x2},\\ y_3=\\var{x3}\\]</p>\n<p>$\\hat{t}=$[[0]]</p>", "gaps": [{"type": "jme", "useCustomName": false, "customName": "", "marks": "2", "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "answer": "{t_est}", "showPreview": true, "checkingType": "dp", "checkingAccuracy": 3, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": false, "singleLetterVariables": false, "allowUnknownFunctions": true, "implicitFunctionComposition": false, "caseSensitive": false, "valuegenerators": []}], "sortAnswers": false}], "partsMode": "all", "maxMarks": 0, "objectives": [], "penalties": [], "objectiveVisibility": "always", "penaltyVisibility": "always", "type": "question"}, {"name": "Method of Moments - Q2", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Tatiana Tyukina", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/341/"}], "tags": [], "metadata": {"description": "", "licence": "None specified"}, "statement": "<p>A random sample of size $n=\\var{n}$ , $\\var{sample}$, is drawn (with replacement) from a population with the following p.m.f.:</p>\n<p>\\[p_X(x)=\\frac{\\var{n}!}{x!(\\var{n}-x)!}\\left(t\\right)^x\\left(1-t\\right)^{\\var{n}-x}, \\quad x=0,1,...,\\var{n}\\]</p>\n<p></p>", "advice": "", "rulesets": {}, "builtin_constants": {"e": true, "pi,\u03c0": true, "i": true}, "constants": [], "variables": {"x1": {"name": "x1", "group": "Ungrouped variables", "definition": "random(0..n)", "description": "", "templateType": "anything", "can_override": false}, "x3": {"name": "x3", "group": "Ungrouped variables", "definition": "random(1..n)", "description": "", "templateType": "anything", "can_override": false}, "x2": {"name": "x2", "group": "Ungrouped variables", "definition": "random(0..n-1)", "description": "", "templateType": "anything", "can_override": false}, "n": {"name": "n", "group": "Ungrouped variables", "definition": "4", "description": "", "templateType": "number", "can_override": false}, "sample": {"name": "sample", "group": "Ungrouped variables", "definition": "\"X1,\\ X2,\\ X3,\\ X4\"", "description": "", "templateType": "anything", "can_override": false}, "x4": {"name": "x4", "group": "Ungrouped variables", "definition": "random(0..n)", "description": "", "templateType": "anything", "can_override": false}}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": ["x1", "x2", "x3", "x4", "sample", "n"], "variable_groups": [], "functions": {}, "preamble": {"js": "", "css": ""}, "parts": [{"type": "gapfill", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<p>Find the estimator for $t$ using the method of moments:</p>\n<p>Find $E[X]$ as a function of $t$:</p>\n<p>$E[X]=$[[0]]</p>\n<p>Find the first sample moment $m_1$ as a function of $\\var{sample}$:</p>\n<p>$m_1=$[[1]]</p>\n<p>Write the estimator for t:</p>\n<p>$\\hat{t}=$[[2]]</p>\n<p></p>", "gaps": [{"type": "jme", "useCustomName": false, "customName": "", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "answer": "{n}*t", "showPreview": true, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": false, "singleLetterVariables": false, "allowUnknownFunctions": true, "implicitFunctionComposition": false, "caseSensitive": false, "valuegenerators": [{"name": "t", "value": ""}]}, {"type": "jme", "useCustomName": false, "customName": "", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "answer": "(X1+X2+X3+X4)/4", "showPreview": true, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": false, "singleLetterVariables": false, "allowUnknownFunctions": true, "implicitFunctionComposition": false, "caseSensitive": false, "valuegenerators": [{"name": "x1", "value": ""}, {"name": "x2", "value": ""}, {"name": "x3", "value": ""}, {"name": "x4", "value": ""}]}, {"type": "jme", "useCustomName": false, "customName": "", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "answer": "(X1+X2+X3+X4)/16", "showPreview": true, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": false, "singleLetterVariables": false, "allowUnknownFunctions": true, "implicitFunctionComposition": false, "caseSensitive": false, "valuegenerators": [{"name": "x1", "value": ""}, {"name": "x2", "value": ""}, {"name": "x3", "value": ""}, {"name": "x4", "value": ""}]}], "sortAnswers": false}, {"type": "gapfill", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<p>Find the estimate of $t$ if&nbsp; the following numbers were observed:</p>\n<p>\\[x1=\\var{x1},\\ x2=\\var{x2},\\ x3=\\var{x3}, \\ x4=\\var{x4}\\]</p>\n<p>$\\hat{t}=$[[0]]</p>", "gaps": [{"type": "jme", "useCustomName": false, "customName": "", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "answer": "({x1}+{x2}+{x3}+{x4})/(4*{n})", "showPreview": true, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": false, "singleLetterVariables": false, "allowUnknownFunctions": true, "implicitFunctionComposition": false, "caseSensitive": false, "valuegenerators": []}], "sortAnswers": false}], "partsMode": "all", "maxMarks": 0, "objectives": [], "penalties": [], "objectiveVisibility": "always", "penaltyVisibility": "always", "type": "question"}, {"name": "Nikon's copy of Log-likelihood and maximum likelihood estimator for PDF", "extensions": ["stats"], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Newcastle University Mathematics and Statistics", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/697/"}, {"name": "Nikon Kurnosov", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/28343/"}], "variable_groups": [], "variables": {"x": {"templateType": "anything", "group": "Ungrouped variables", "definition": "x1+x2+x3", "description": "", "name": "x"}, "tol": {"templateType": "anything", "group": "Ungrouped variables", "definition": "0", "description": "", "name": "tol"}, "x1": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(2.5..5.5#0.5)", "description": "", "name": "x1"}, "prod": {"templateType": "anything", "group": "Ungrouped variables", "definition": "x1*x2*x3", "description": "", "name": "prod"}, "m": {"templateType": "anything", "group": "Ungrouped variables", "definition": "-precround(3/mle^2,2)", "description": "", "name": "m"}, "where": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random('North Shields','Owslebury','Bradfield in West Yorkshire','Sheffield','Windy Nook', 'Hepple','Leeming','Linton-on-Ouse','Scampton','Cranwell','Keele','Bingley', 'Ecclefechan','Finningley','Foxrock, Dublin')", "description": "", "name": "where"}, "x2": {"templateType": "anything", "group": "Ungrouped variables", "definition": "if(y2=x1,y2+ random(-1..1#0.5),y2)", "description": "", "name": "x2"}, "y2": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(3.5..6.5#0.5)", "description": "", "name": "y2"}, "sumsq": {"templateType": "anything", "group": "Ungrouped variables", "definition": "x1^2+x2^2+x3^2", "description": "", "name": "sumsq"}, "x3": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(2.5..7.5#0.5)", "description": "", "name": "x3"}, "mle": {"templateType": "anything", "group": "Ungrouped variables", "definition": "precround(3/sumsq,2)", "description": "", "name": "mle"}}, "ungrouped_variables": ["prod", "mle", "m", "x1", "sumsq", "tol", "x3", "x2", "x", "y2", "where"], "rulesets": {"std": ["all", "fractionNumbers", "!collectNumbers", "!noLeadingMinus"]}, "functions": {}, "showQuestionGroupNames": false, "parts": [{"scripts": {}, "gaps": [{"answer": "({(8 *prod)} * (t ^ 3) * Exp(({( - sumsq)} * t)))", "vsetrange": [-0.2, -0.1], "scripts": {}, "checkvariablenames": false, "expectedvariablenames": [], "showpreview": true, "checkingtype": "absdiff", "checkingaccuracy": 0.001, "answersimplification": "std", "type": "jme", "showCorrectAnswer": true, "marks": 1, "vsetrangepoints": 5}], "type": "gapfill", "showCorrectAnswer": true, "prompt": "\n      \n      \n      <p>Find the likelihood function for $t$ given these observations.</p>\n      \n      \n      \n      <p>$L(t|\\underline{x})=\\;\\;$[[0]]</p>\n      \n      ", "marks": 0}, {"scripts": {}, "gaps": [{"answer": "(Ln({(8 * prod)}) + (3 * Ln(t))  - ({sumsq} * t))", "vsetrange": [0, 1], "scripts": {}, "checkvariablenames": false, "expectedvariablenames": [], "showpreview": true, "checkingtype": "absdiff", "checkingaccuracy": 0.001, "answersimplification": "std", "type": "jme", "showCorrectAnswer": true, "marks": 1, "vsetrangepoints": 5}], "type": "gapfill", "showCorrectAnswer": true, "prompt": "<p>Hence find the log-likelihood function for $t$</p>\n        <p>$l(t|\\underline{x})=\\;\\;$[[0]]</p>\n        <p>If $\\ln(a)$, for some integer $a$, is a term in your answer, leave as $\\ln(a)$ and do not evaluate.</p>", "marks": 0}, {"scripts": {}, "gaps": [{"showPrecisionHint": false, "scripts": {}, "allowFractions": false, "type": "numberentry", "correctAnswerFraction": false, "minValue": "mle-tol", "showCorrectAnswer": true, "marks": 1, "maxValue": "mle+tol"}], "type": "gapfill", "showCorrectAnswer": true, "prompt": "<p>Find the <span class=\"caps\">MLE</span> $\\hat{t}$ for $t$</p>\n        <p>$\\hat{t}=\\;\\;$[[0]]</p>\n        <p>Input to 2 decimal places.</p>", "marks": 0}, {"scripts": {}, "gaps": [{"answer": "-(3/t^2)", "vsetrange": [0, 1], "scripts": {}, "checkvariablenames": false, "expectedvariablenames": [], "showpreview": true, "checkingtype": "absdiff", "checkingaccuracy": 0.001, "answersimplification": "std", "type": "jme", "showCorrectAnswer": true, "marks": 1, "vsetrangepoints": 5}, {"showPrecisionHint": false, "scripts": {}, "allowFractions": false, "type": "numberentry", "correctAnswerFraction": false, "minValue": "m-tol", "showCorrectAnswer": true, "marks": 1, "maxValue": "m+tol"}], "type": "gapfill", "showCorrectAnswer": true, "prompt": "<p>Now verify that you have indeed found a maximum:</p>\n        <p>1. First find $\\displaystyle\\frac{\\partial^2\\;l}{\\partial\\;t^2}=\\;\\;$[[0]].</p>\n        <p>2. Using the value of the MLE to 2 decimal places you have found:&nbsp;</p>\n        <p>$\\displaystyle\\frac{\\partial^2\\;l}{\\partial\\;t^2}$ evaluated at $\\hat{t}$&nbsp;= [[1]].</p>\n        <p>Input to 2 decimal places.</p>", "marks": 0}], "statement": "<p>The average annual wind speed, $X$, at {where} has the following probability density function with parameter $t$ which you have to estimate:</p>\n    <table>\n    <tbody>\n    <tr>\n    <td rowspan=\"3\">$f_X(x) = \\left \\{ \\begin{array}{l} \\phantom{{.}} \\\\ \\phantom{{.}} \\\\ \\phantom{{.}} \\end{array} \\right .$</td>\n    <td style=\"text-align: left;\">$2txe^{-tx^2}$</td>\n    <td>$x \\gt 0,$</td>\n    </tr>\n    <tr>\n    <td>&nbsp;</td>\n    <td>&nbsp;</td>\n    </tr>\n    <tr>\n    <td style=\"text-align: left;\">$0$</td>\n    <td>$\\textrm{otherwise.}$</td>\n    </tr>\n    </tbody>\n    </table>\n    <p>For three randomly selected years, we observe the following average wind speeds:</p>\n    <p>$x_1=\\var{x1},\\;\\;x_2=\\var{x2}$ and $x_3=\\var{x3}$.</p>", "tags": ["checked2015", "cr1", "density function", "estimators", "likelihood functions", "log-likelihood function", "maximum", "maximum likelihood estimator", "mle", "MLE", "PDF", "pdf", "Probability", "probability", "probability density function", "random sample", "random variable", "sc", "second derivative", "statistics", "tested1", "unused"], "question_groups": [{"pickingStrategy": "all-ordered", "questions": [], "name": "", "pickQuestions": 0}], "preamble": {"css": "", "js": ""}, "type": "question", "metadata": {"notes": "<p><strong>14/07/2012:</strong></p>\n        <p>Added tags.</p>\n        <p>Corrected mistakes in Advice.</p>\n        <p>Added some text to make statement clearer re parameter $t$.</p>\n        <p>Rephrased questions in last question so that it is clear that the value to 2dps is used in the calculation.</p>\n        <p>Spaced Advice text.</p>\n        <p>New tolerance variable, &nbsp;tol=0 for last two questions.</p>\n        <p>Added line in prompt: If $\\ln(a)$, for some integer $a$, is a term in your answer, leave as $\\ln(a)$ and do not evaluate.</p>\n        <p>Improved display of correct answer in second question as $+\\;- $ together. Also improved correct answer display in second last question.</p>\n        <p>Important: set checking range between -0.2 and -0.1 rather than between 0 and 1 so that evaluation of likelihood function over the range does not suffer from underflow and incorrect answer marked as correct. This needs constant testing, have tested on bounday values and OK.</p>\n        <p><strong>1/08/2012:</strong></p>\n        <p>Added tags.</p>\n        <p>In the Advice section, moved \\Rightarrow to the beginning of the line instead of the end of the previous line.</p>\n        <p><strong>21/12/2012:</strong></p>\n        <p>Checked calculations, OK. Added tested1 tag.</p>\n        <p>Checked rounding, OK. Added cr1 tag.</p>\n        <p>Scenarios, so added sc tag.</p>", "licence": "Creative Commons Attribution 4.0 International", "description": "<p>Given a PDF $f(x)$ on the real line with unknown parameter $t$ and three random observations, find log-likelihood and MLE $\\hat{t}$ for $t$.&nbsp;</p>"}, "variablesTest": {"condition": "", "maxRuns": 100}, "advice": "<p>a)</p>\n    <p>\\[ \\begin{eqnarray*} L(t|\\underline{x})&amp;=&amp; \\var{2*x1}te^{-\\var{x1}^2t}\\times \\var{2*x2}te^{-\\var{x2}^2t} \\times \\var{2*x3}te^{-\\var{x3}^2t}\\\\ &amp;=&amp; \\var{8*prod}t^3e^{-\\var{sumsq}t} \\end{eqnarray*} \\]<br/>b)</p>\n    <p>The log-likelihood function is :<br/>\\[\\begin{eqnarray*} l(t|\\underline{x})&amp;=&amp;\\ln\\left( \\var{8*prod}t^3e^{-\\var{sumsq}t}\\right)\\\\ &amp;=&amp;\\ln(\\var{8*prod})+3\\ln(t)-\\var{sumsq}t \\end{eqnarray*} \\]</p>\n    <p>c)</p>\n    <p>We have:<br/>\\[\\frac{\\partial\\;l}{\\partial\\;t}=\\frac{3}{t}-\\var{sumsq}\\]<br/>Now:<br/>\\[\\begin{eqnarray*} \\frac{\\partial\\;l}{\\partial\\;t}&amp;=&amp;0 \\\\ \\Rightarrow \\frac{3}{t}-\\var{sumsq}&amp;=&amp;0\\\\ \\Rightarrow t&amp;=&amp;\\frac{3}{\\var{sumsq}} = \\var{mle} \\end{eqnarray*} \\] to 3 decimal places.<br/>And putting $t=\\hat{t}$ gives the <span class=\"caps\">MLE</span> $\\hat{t}=\\var{mle}$</p>\n    <p>d)<br/>\\[\\frac{\\partial^2\\;l}{\\partial\\;t^2}=-3t^{-2} \\lt 0\\]<br/>when evaluated at any point including $t=\\hat{t}=\\var{mle}$.</p>\n    <p>Hence gives a maximum at $t=\\hat{t}$.</p>"}, {"name": "Find maximum likelihood estimator of mean of normal distribution", "extensions": ["stats"], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Newcastle University Mathematics and Statistics", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/697/"}], "variable_groups": [], "variables": {"w": {"group": "Ungrouped variables", "templateType": "anything", "definition": "if(r=4,1,0)", "name": "w", "description": ""}, "mlephi": {"group": "Ungrouped variables", "templateType": "anything", "definition": "precround(( t * ln(m/n+ b) + u * (a / (m/n + d)) + v * e ^ (m/(n*c)) + w * ((m/n) ^ 2 + g)),2)", "name": "mlephi", "description": ""}, "v": {"group": "Ungrouped variables", "templateType": "anything", "definition": "if(r=3,1,0)", "name": "v", "description": ""}, "b": {"group": "Ungrouped variables", "templateType": "anything", "definition": "random(1..9)", "name": "b", "description": ""}, "c": {"group": "Ungrouped variables", "templateType": "anything", "definition": "random(10..15)", "name": "c", "description": ""}, "var": {"group": "Ungrouped variables", "templateType": "anything", "definition": "random(10..30)", "name": "var", "description": ""}, "t": {"group": "Ungrouped variables", "templateType": "anything", "definition": "if(r=1,1,0)", "name": "t", "description": ""}, "a": {"group": "Ungrouped variables", "templateType": "anything", "definition": "random(2..9)", "name": "a", "description": ""}, "tol": {"group": "Ungrouped variables", "templateType": "anything", "definition": "0", "name": "tol", "description": ""}, "mlemu": {"group": "Ungrouped variables", "templateType": "anything", "definition": "precround(m/n,2)", "name": "mlemu", "description": ""}, "r": {"group": "Ungrouped variables", "templateType": "anything", "definition": "random(1..4)", "name": "r", "description": ""}, "n": {"group": "Ungrouped variables", "templateType": "anything", "definition": "random(15..30)", "name": "n", "description": ""}, "d": {"group": "Ungrouped variables", "templateType": "anything", "definition": "random(2..9)", "name": "d", "description": ""}, "g": {"group": "Ungrouped variables", "templateType": "anything", "definition": "random(2..9)", "name": "g", "description": ""}, "u": {"group": "Ungrouped variables", "templateType": "anything", "definition": "if(r=2,1,0)", "name": "u", "description": ""}, "m": {"group": "Ungrouped variables", "templateType": "anything", "definition": "random(150..500#5)", "name": "m", "description": ""}}, "ungrouped_variables": ["a", "c", "b", "d", "g", "m", "n", "w", "mlemu", "u", "t", "tol", "v", "var", "r", "mlephi"], "rulesets": {"std": ["all", "fractionNumbers", "!collectNumbers", "!noLeadingMinus"]}, "functions": {}, "showQuestionGroupNames": false, "parts": [{"scripts": {}, "gaps": [{"correctAnswerFraction": false, "showPrecisionHint": false, "allowFractions": false, "scripts": {}, "type": "numberentry", "maxValue": "mlemu+tol", "minValue": "mlemu-tol", "showCorrectAnswer": true, "marks": 1}], "type": "gapfill", "prompt": "\n      \n      \n      <p>Using this data find the <span class=\"caps\">MLE</span> $\\hat{\\mu}$ of $\\mu$ .</p>\n      \n      \n      \n      <p>$\\hat{\\mu}=\\;\\;$[[0]]</p>\n      \n      \n      \n      <p>Enter to 2 decimal places.</p>\n      \n      \n        ", "showCorrectAnswer": true, "marks": 0}, {"scripts": {}, "gaps": [{"correctAnswerFraction": false, "showPrecisionHint": false, "allowFractions": false, "scripts": {}, "type": "numberentry", "maxValue": "mlephi+tol", "minValue": "mlephi-tol", "showCorrectAnswer": true, "marks": 1}], "type": "gapfill", "prompt": "<p>Let \\[\\phi = \\simplify[std]{{t}*ln(mu+{b})+{u}*({a}/(mu+{d}))+{v}e^(mu/{c})+{w}*(mu^2+{g})}\\]</p>\n<p>Using the invariance property of maximum likelihood estimators, what is the MLE $\\hat{\\phi}$ of $\\phi$?</p>\n<p>$\\hat{\\phi}=\\;\\;$[[0]] (make sure that you use an estimator for $\\mu$ which is accurate to 4 decimal places).</p>\n<p>Enter to 2 decimal places.</p>", "showCorrectAnswer": true, "marks": 0}], "statement": "\n    <p>In an experiment we take $\\var{n}$ observations $x_i$ from a Normal distribution with unknown mean $\\mu$ and variance $\\var{var}$.</p>\n  <p>From this sample we find:</p>\n  <p>\\[\\sum_{i=1}^{\\var{n}}x_i = \\var{m}\\]</p>\n    ", "tags": ["checked2015", "cr1", "distributions", "functions", "invariance property of maximum likelihood estimators", "known variance", "MAS2302", "maximum likelihood estimator", "mean", "mean ", "mle", "MLE", "Normal distribution", "normal distribution", "query", "sample", "statistics", "sum of sample", "tested1", "unknown mean", "variance"], "question_groups": [{"pickingStrategy": "all-ordered", "questions": [], "name": "", "pickQuestions": 0}], "preamble": {"css": "", "js": ""}, "type": "question", "metadata": {"notes": "<p><strong>14/07/2012:</strong></p>\n        <p>Added tags.</p>\n        <p>New tolerance variable t=0 for the numeric inputs.</p>\n        <p>Checked calculations.</p>\n        <p><strong>1/08/2012:</strong></p>\n        <p>Added tags.</p>\n        <p>Question appears to be working correctly.</p>\n        <p><strong>21/12/2012:</strong></p>\n        <p><strong><br /></strong>Checked calculations for all functions appearing in this exercise. Added tested1 tag.</p>\n        <p>Rounding OK, added cr1 tag.</p>\n        <p>In order to calculate the MLE for $\\phi$ need to use a more accurate value than that found for $\\mu$ to 2 decimal places. This is stated in the question now.</p>\n        <p>There is no need to have the variance given. Added query tag for that.&nbsp;</p>\n        <p>Also query the accuracy needed for the estimator of $\\mu$.</p>", "licence": "Creative Commons Attribution 4.0 International", "description": "<p>Given sum of sample from a Normal distribution with unknown mean $\\mu$ and known variance $\\sigma^2$. Find MLE of $\\mu$ and one of four functions of $\\mu$.</p>"}, "variablesTest": {"condition": "", "maxRuns": 100}, "advice": "<p>a)<br/>The <span class=\"caps\">MLE</span> $\\hat{\\mu}$ of $\\mu$ is given by dividing the sum of the sample by the number of samples i.e.</p>\n<p>\\[\\hat{\\mu}=\\frac{\\var{m}}{\\var{n}}=\\var{mlemu}\\] to 2 decimal places.</p>\n<p>b)<br/>In order to find the <span class=\"caps\">MLE</span> $\\hat{\\phi}$ of \\[\\phi=\\simplify[std]{{t}*ln(mu+{b})+{u}*({a}/(mu+{d}))+{v}e^(mu/{c})+{w}*(mu^2+{g})}\\]</p>\n<p>we note that $\\phi$ is 1-1 and so we can directly substitute $\\mu=\\hat{\\mu}$ into this to obtain:</p>\n<p>\\[\\hat{\\phi} = \\simplify[std]{{t} * Ln({m} / {n} + {b}) + {u} * ({a} / ({m} / {n} + {d})) + {v} * e ^ (({m} / {n}) / {c}) + {w} * (({m} / {n}) ^ 2 + {g})} = \\var{mlephi}\\]</p>\n<p>to 2 decimal places.</p>"}, {"name": "Nikon's copy of 20122013 CBA1_1", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Newcastle University Mathematics and Statistics", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/697/"}, {"name": "Nikon Kurnosov", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/28343/"}], "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}], "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}, {"name": "Nikon's copy of Expectation and variance of combinations of estimators", "extensions": ["stats"], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Newcastle University Mathematics and Statistics", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/697/"}, {"name": "Nikon Kurnosov", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/28343/"}], "variable_groups": [], "variables": {"b": {"templateType": "anything", "group": "Ungrouped variables", "definition": "'B'", "name": "b", "description": ""}, "e3": {"templateType": "anything", "group": "Ungrouped variables", "definition": "s3*m", "name": "e3", "description": ""}, "unb2": {"templateType": "anything", "group": "Ungrouped variables", "definition": "if(unb1=0,1,random(0,1))", "name": "unb2", "description": ""}, "s7": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(1,-1)", "name": "s7", "description": ""}, "sqsum1": {"templateType": "anything", "group": "Ungrouped variables", "definition": "cw1^2+cx1^2+cy1^2+cz1^2", "name": "sqsum1", "description": ""}, "wrong": {"templateType": "anything", "group": "Ungrouped variables", "definition": "if(unb1*unb2>0,'C',if(unb1=1,'B','A'))", "name": "wrong", "description": ""}, "t3": {"templateType": "anything", "group": "Ungrouped variables", "definition": "if(unb3=1,1,random(2..9))", "name": "t3", "description": ""}, "v1": {"templateType": "anything", "group": "Ungrouped variables", "definition": "sqsum1*sd^2", "name": "v1", "description": ""}, "su2": {"templateType": "anything", "group": "Ungrouped variables", "definition": "cw2+cx2+cy2+cz2", "name": "su2", "description": ""}, "p": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random('W','X','Y','Z')", "name": "p", "description": ""}, "cy1": {"templateType": "anything", "group": "Ungrouped variables", "definition": "if(cx1=0,random(1..9),random(-9..9))", "name": "cy1", "description": ""}, "sd": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(2..6)", "name": "sd", "description": ""}, "cw1": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(1..9)", "name": "cw1", "description": ""}, "t1": {"templateType": "anything", "group": "Ungrouped variables", "definition": "if(unb1=1,1,random(2..9))", "name": "t1", "description": ""}, "tw": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(0,1)", "name": "tw", "description": ""}, "cy3": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(2..9)", "name": "cy3", "description": ""}, "sqsum2": {"templateType": "anything", "group": "Ungrouped variables", "definition": "cw2^2+cx2^2+cy2^2+cz2^2", "name": "sqsum2", "description": ""}, "v3": {"templateType": "anything", "group": "Ungrouped variables", "definition": "sqsum3*sd^2", "name": "v3", "description": ""}, "s1": {"templateType": "anything", "group": "Ungrouped variables", "definition": "cw1+cx1+cy1+cz1", "name": "s1", "description": ""}, "tol": {"templateType": "anything", "group": "Ungrouped variables", "definition": "0.01", "name": "tol", "description": ""}, "a": {"templateType": "anything", "group": "Ungrouped variables", "definition": "'A'", "name": "a", "description": ""}, "c": {"templateType": "anything", "group": "Ungrouped variables", "definition": "'C'", "name": "c", "description": ""}, "sqsum3": {"templateType": "anything", "group": "Ungrouped variables", "definition": "cw3^2+cx3^2+cy3^2+cz3^2", "name": "sqsum3", "description": ""}, "s2": {"templateType": "anything", "group": "Ungrouped variables", "definition": "if(unb2=1,su2,su2+random(1..4))", "name": "s2", "description": ""}, "unb1": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(0,1)", "name": "unb1", "description": ""}, "cz3": {"templateType": "anything", "group": "Ungrouped variables", "definition": "t3-cw3-cx3-cy3", "name": "cz3", "description": ""}, "cx2": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(1..9)", "name": "cx2", "description": ""}, "cw3": {"templateType": "anything", "group": "Ungrouped variables", "definition": "tw*random(10..40)", "name": "cw3", "description": ""}, "sx": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(1,-1)", "name": "sx", "description": ""}, "correct2": {"templateType": "anything", "group": "Ungrouped variables", "definition": "if(unb1=1,if(unb2=1,'B','C'),'C')", "name": "correct2", "description": ""}, "cw2": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(1..9)", "name": "cw2", "description": ""}, "q": {"templateType": "anything", "group": "Ungrouped variables", "definition": "switch(p='X',random('W','Y','Z'),p='W',random('X','Y','Z'),p='Y',random('W','X','Z'),random('W','X','Y'))", "name": "q", "description": ""}, "e2": {"templateType": "anything", "group": "Ungrouped variables", "definition": "precround(su2*m/S2,2)", "name": "e2", "description": ""}, "e1": {"templateType": "anything", "group": "Ungrouped variables", "definition": "s1*m", "name": "e1", "description": ""}, "m": {"templateType": "anything", "group": "Ungrouped variables", "definition": "s7*random(1..10)", "name": "m", "description": ""}, "correct1": {"templateType": "anything", "group": "Ungrouped variables", "definition": "if(unb1=1,'A','B')", "name": "correct1", "description": ""}, "cz2": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(1..9)", "name": "cz2", "description": ""}, "unb3": {"templateType": "anything", "group": "Ungrouped variables", "definition": "if(unb1*unb2>0,0,1)", "name": "unb3", "description": ""}, "cy2": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(1..9)", "name": "cy2", "description": ""}, "cx1": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(-9..9)", "name": "cx1", "description": ""}, "s3": {"templateType": "anything", "group": "Ungrouped variables", "definition": "cw3+cx3+cy3+cz3", "name": "s3", "description": ""}, "v2": {"templateType": "anything", "group": "Ungrouped variables", "definition": "precround(sqsum2*sd^2/S2^2,2)", "name": "v2", "description": ""}, "cx3": {"templateType": "anything", "group": "Ungrouped variables", "definition": "tx*sx*random(1..9)", "name": "cx3", "description": ""}, "cz1": {"templateType": "anything", "group": "Ungrouped variables", "definition": "t1-cx1-cw1-cy1", "name": "cz1", "description": ""}, "tx": {"templateType": "anything", "group": "Ungrouped variables", "definition": "if(tw=0,1,0)", "name": "tx", "description": ""}}, "ungrouped_variables": ["cy3", "cy2", "cy1", "t3", "correct2", "correct1", "t1", "unb3", "unb2", "unb1", "s3", "tw", "s1", "s7", "tol", "cx1", "cx2", "cx3", "sqsum1", "sqsum3", "sqsum2", "cz2", "cz3", "cz1", "v1", "v2", "v3", "e1", "e3", "e2", "a", "c", "b", "tx", "cw1", "cw3", "cw2", "m", "wrong", "sx", "q", "p", "su2", "s2", "sd"], "question_groups": [{"pickingStrategy": "all-ordered", "questions": [], "name": "", "pickQuestions": 0}], "showQuestionGroupNames": false, "functions": {}, "parts": [{"scripts": {}, "gaps": [{"showPrecisionHint": false, "scripts": {}, "allowFractions": false, "type": "numberentry", "showCorrectAnswer": true, "minValue": "{e1}", "correctAnswerFraction": false, "marks": 1, "maxValue": "{e1}"}, {"showPrecisionHint": false, "scripts": {}, "allowFractions": false, "type": "numberentry", "showCorrectAnswer": true, "minValue": "{v1}", "correctAnswerFraction": false, "marks": 1, "maxValue": "{v1}"}], "type": "gapfill", "showCorrectAnswer": true, "prompt": "<p>$\\simplify[std]{A = {cw1} * W + {cx1} * X + {cy1} * Y + {cz1} * Z}$</p>\n      <p>$ \\operatorname{E}[A]=\\;\\;$[[0]]$\\;\\;\\;\\;\\operatorname{Var}(A)=\\;\\;\\;$[[1]]</p>\n      <p>Input both as integers.</p>", "marks": 0}, {"scripts": {}, "gaps": [{"showPrecisionHint": false, "scripts": {}, "allowFractions": false, "type": "numberentry", "showCorrectAnswer": true, "minValue": "{e2-tol}", "correctAnswerFraction": false, "marks": 1, "maxValue": "{e2+tol}"}, {"showPrecisionHint": false, "scripts": {}, "allowFractions": false, "type": "numberentry", "showCorrectAnswer": true, "minValue": "{v2-tol}", "correctAnswerFraction": false, "marks": 1, "maxValue": "{v2+tol}"}], "type": "gapfill", "showCorrectAnswer": true, "prompt": "<p>$\\simplify[std]{B = (1 / {S2}) * ({cw2} * W + {cx2} * X + {cy2} * Y + {cz2} * Z)}$</p>\n      <p>$\\operatorname{E}[B]=\\;\\;$[[0]]$\\;\\;\\;\\;\\operatorname{Var}(B)=\\;\\;\\;$[[1]]</p>\n      <p>Input both to 2 decimal places.</p>", "marks": 0}, {"scripts": {}, "gaps": [{"showPrecisionHint": false, "scripts": {}, "allowFractions": false, "type": "numberentry", "showCorrectAnswer": true, "minValue": "{e3}", "correctAnswerFraction": false, "marks": 1, "maxValue": "{e3}"}, {"showPrecisionHint": false, "scripts": {}, "allowFractions": false, "type": "numberentry", "showCorrectAnswer": true, "minValue": "{v3}", "correctAnswerFraction": false, "marks": 1, "maxValue": "{v3}"}], "type": "gapfill", "showCorrectAnswer": true, "prompt": "<p>$\\simplify[std]{C = {cw3} * W + {cx3} * X + {cy3} * Y + {cz3} * Z}$</p>\n      <p>$ \\operatorname{E}[C]=\\;\\;$[[0]]$\\;\\;\\;\\;\\operatorname{Var}(C)=\\;\\;\\;$[[1]]</p>\n      <p>Input both as integers.</p>", "marks": 0}, {"scripts": {}, "gaps": [{"displayType": "checkbox", "choices": ["<p>$\\var{Correct1}$</p>", "<p>$\\var{Correct2}$</p>", "<p>$\\var{Wrong}$</p>"], "matrix": [1, 1, -1], "distractors": ["", "", ""], "minAnswers": 0, "maxAnswers": 0, "shuffleChoices": true, "scripts": {}, "maxMarks": 0, "type": "m_n_2", "minMarks": 0, "showCorrectAnswer": true, "displayColumns": 3, "marks": 0}], "type": "gapfill", "showCorrectAnswer": true, "prompt": "\n      \n      \n      <p>Which of the estimators $A,\\;\\;B$ or $C$ above are unbiased for $\\mu$? Select the correct choices. <br/>You will lose a mark for selecting a wrong choice.</p>\n      \n      \n      \n      <p>[[0]]</p>\n      \n      ", "marks": 0}, {"scripts": {}, "gaps": [{"displayType": "radiogroup", "choices": ["<p>$\\var{B}$</p>", "<p>$\\var{A}$</p>", "<p>$\\var{C}$</p>"], "matrix": [1, 0, 0], "distractors": ["", "", ""], "shuffleChoices": true, "scripts": {}, "maxMarks": 0, "type": "1_n_2", "minMarks": 0, "showCorrectAnswer": true, "displayColumns": 3, "marks": 0}], "type": "gapfill", "showCorrectAnswer": true, "prompt": "\n      \n      \n      <p>Which of the estimators $A,\\;\\;B$ or $C$ above is the most efficient?</p>\n      \n      \n      \n      <p>[[0]]</p>\n      \n      ", "marks": 0}, {"scripts": {}, "gaps": [{"showPrecisionHint": false, "scripts": {}, "allowFractions": false, "type": "numberentry", "showCorrectAnswer": true, "minValue": "{m^2}", "correctAnswerFraction": false, "marks": 1, "maxValue": "{m^2}"}], "type": "gapfill", "showCorrectAnswer": true, "prompt": "<p>Find $\\operatorname{E}[\\var{p}\\var{q}]=\\;\\;$[[0]]</p>", "marks": 0}], "variablesTest": {"condition": "", "maxRuns": 100}, "statement": "<p>Suppose $W,\\;\\;X,\\;\\;Y,\\;\\;$ and $Z$ are i.i.d. variables with mean $\\mu=\\var{m}$, standard deviation $\\sigma=\\var{sd}$</p>\n    <p>Find the expectation and variance of each of the following estimators of $\\mu$.</p>", "tags": ["IID", "MAS1604", "biased", "checked2015", "cr1", "efficient estimators", "estimators", "expectation", "i.i.d", "identical independent distributions", "iid", "independent identical distributions", "mean ", "random variables", "standard deviation", "statistics", "tested1", "unbiased", "unbiased estimators", "variance"], "rulesets": {"std": ["all", "fractionNumbers", "!collectNumbers", "!noLeadingMinus"]}, "preamble": {"css": "", "js": ""}, "type": "question", "metadata": {"notes": "<p><strong>13/07/2012:</strong></p>\n            <p>Added tags.</p>\n            <p>Improved and made consistent the display in various content areas.</p>\n            <p>Set new tolerance variable tol=0 for 2 dps numeric input questions.</p>\n            <p>Added formula for $\\operatorname{Var}(aR+bS)$.</p>\n            <p>Checked calculation.</p>\n            <p>Added description.</p>\n            <p><strong>1/08/2012:</strong></p>\n            <p>Added tags.</p>\n            <p>Question appears to be working correctly.</p>\n            <p><strong>21/12/2012:</strong></p>\n            <p>Checked calculation, OK. Added tested1 tag.</p>\n            <p>Checked rounding, OK. Added tag cr1.</p>", "licence": "Creative Commons Attribution 4.0 International", "description": "<p>Given three linear combinations of&nbsp;four i.i.d. variables, find the expectation and variance of these estimators of the mean $\\mu$. Which are unbiased and efficient?</p>"}, "advice": "<p>We use the result throughout these solutions that for independent random variables $R$ and $S$ then $\\operatorname{E}[aR+bS]=a \\operatorname{E}[R]+b\\operatorname{E}[S]$ and $\\operatorname{Var}(aR+bS)=a^2\\operatorname{Var}(R)+b^2\\operatorname{Var}(S)$.</p>\n  <p>a)<br/>\\[\\begin{eqnarray*} \\operatorname{E}[A] &amp;=&amp; \\operatorname{E}[\\simplify[std]{{cw1} * W + {cx1} * X + {cy1} * Y + {cz1} * Z}]\\\\ &amp;=&amp; \\simplify[std]{{cw1} * {m} + {cx1} * {m} + {cy1} * {m} + {cz1} * {m}}\\\\ &amp;=&amp; \\var{e1}\\\\ \\\\ \\\\ \\operatorname{Var}(A) &amp;=&amp; \\operatorname{Var}[\\simplify[std]{{cw1} * W + {cx1} * X + {cy1} * Y + {cz1} * Z}]\\\\ &amp;=&amp; \\simplify[std]{ {cw1 ^ 2} * Var(W) + {cx1 ^ 2} * Var(X) + {cy1 ^ 2} * Var(Y) + {cz1 ^ 2} * Var(Z)}\\\\ &amp;=&amp; \\simplify[std]{{cw1 ^ 2} * {sd ^ 2} + {cx1 ^ 2} * {sd ^ 2} + {cy1 ^ 2} * {sd ^ 2} + {cz1 ^ 2} * {sd ^ 2} }\\\\ &amp;=&amp; \\var{v1} \\end{eqnarray*} \\]</p>\n  <p>b)<br/>\\[\\begin{eqnarray*} \\operatorname{E}[B] &amp;=&amp; \\frac{1}{\\var{S2}}\\left(\\operatorname{E}[\\simplify[std]{{cw2} * W + {cx2} * X + {cy2} * Y + {cz2} * Z}]\\right)\\\\ &amp;=&amp; \\frac{1}{\\var{S2}}\\left(\\simplify[std]{{cw2} * {m} + {cx2} * {m} + {cy2} * {m} + {cz2} * {m}}\\right)\\\\ &amp;=&amp; \\var{e2}\\\\ \\\\ \\\\ \\operatorname{Var}(B) &amp;=&amp; \\frac{1}{\\var{S2^2}}\\left(\\operatorname{Var}[\\simplify[std]{{cw2} * W + {cx2} * X + {cy2} * Y + {cz2} * Z}]\\right)\\\\ &amp;=&amp; \\frac{1}{\\var{S2^2}}\\left(\\simplify[std]{ {cw2 ^ 2} * Var(W) + {cx2 ^ 2} * Var(X) + {cy2 ^ 2} * Var(Y) + {cz2 ^ 2} * Var(Z)}\\right)\\\\ &amp;=&amp; \\frac{1}{\\var{S2^2}}\\left(\\simplify[std]{{cw2 ^ 2} * {sd ^ 2} + {cx2 ^ 2} * {sd ^ 2} + {cy2 ^ 2} * {sd ^ 2} + {cz2 ^ 2} * {sd ^ 2} }\\right)\\\\ &amp;=&amp; \\var{v2} \\end{eqnarray*} \\]<br/>c)<br/>\\[\\begin{eqnarray*} \\operatorname{E}[C] &amp;=&amp; \\operatorname{E}[\\simplify[std,collectNumbers]{{cw3} * W + {cx3} * X + {cy3} * Y + {cz3} * Z}]\\\\ &amp;=&amp; \\simplify[std,collectNumbers]{{cw3} * {m} + {cx3} * {m} + {cy3} * {m} + {cz3} * {m}}\\\\ &amp;=&amp; \\var{e3}\\\\ \\\\ \\\\ \\operatorname{Var}(C) &amp;=&amp; \\operatorname{Var}[\\simplify[std]{{cw3} * W + {cx3} * X + {cy3} * Y + {cz3} * Z}]\\\\ &amp;=&amp; \\simplify[std]{ {cw3 ^ 2} * Var(W) + {cx3 ^ 2} * Var(X) + {cy3 ^ 2} * Var(Y) + {cz3 ^ 2} * Var(Z)}\\\\ &amp;=&amp; \\simplify[std]{{cw3 ^ 2} * {sd ^ 2} + {cx3 ^ 2} * {sd ^ 2} + {cy3 ^ 2} * {sd ^ 2} + {cz3 ^ 2} * {sd ^ 2} }\\\\ &amp;=&amp; \\var{v3} \\end{eqnarray*} \\]</p>\n  <p>d)</p>\n  <p>We see that $\\var{Correct1},\\;\\;\\var{Correct2}\\;\\;$ are unbiased estimators for $\\mu=\\var{m}$ as their expectations are $\\var{m}$.</p>\n  <p>e)</p>\n  <p>The most efficient estimator is $B$ as it has the smallest variance.</p>\n  <p>f)<br/>Since $\\var{p}$ and $\\var{q}$ are independent we have:</p>\n  <p>$\\operatorname{E}[\\var{p}\\var{q}]=\\operatorname{E}[\\var{p}]\\operatorname{E}[\\var{q}] = \\var{m}\\times \\var{m} = \\var{m^2}$</p>"}]}], "allowPrinting": true, "navigation": {"allowregen": false, "reverse": true, "browse": true, "allowsteps": true, "showfrontpage": true, "navigatemode": "sequence", "onleave": {"action": "none", "message": ""}, "preventleave": true, "typeendtoleave": false, "startpassword": "", "autoSubmit": true, "allowAttemptDownload": false, "downloadEncryptionKey": "", "showresultspage": "review"}, "timing": {"allowPause": true, "timeout": {"action": "none", "message": ""}, "timedwarning": {"action": "none", "message": ""}}, "feedback": {"enterreviewmodeimmediately": false, "showactualmarkwhen": "inreview", "showtotalmarkwhen": "always", "showanswerstatewhen": "inreview", "showpartfeedbackmessageswhen": "never", "showexpectedanswerswhen": "inreview", "showadvicewhen": "inreview", "allowrevealanswer": false, "intro": "", "end_message": "", "results_options": {"printquestions": false, "printadvice": false}, "feedbackmessages": [], "reviewshowexpectedanswer": true, "showanswerstate": false, "reviewshowfeedback": true, "showactualmark": false, "showtotalmark": true, "reviewshowscore": true, "reviewshowadvice": true}, "diagnostic": {"knowledge_graph": {"topics": [], "learning_objectives": []}, "script": "diagnosys", "customScript": ""}, "contributors": [{"name": "Nikon Kurnosov", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/28343/"}], "extensions": ["stats"], "custom_part_types": [], "resources": []}