// Numbas version: finer_feedback_settings
{"name": "Unbiased estimator for exponential distribution", "extensions": ["stats"], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"variable_groups": [], "variables": {"where": {"group": "Ungrouped variables", "templateType": "anything", "definition": "random('at a supermarket checkout','at a fish shop queue','of calls to a call centre','of buses at a station','of hits at a particular website')", "name": "where", "description": ""}, "tol": {"group": "Ungrouped variables", "templateType": "anything", "definition": "0", "name": "tol", "description": ""}, "t": {"group": "Ungrouped variables", "templateType": "anything", "definition": "precround((1-1/n)*1/m,2)", "name": "t", "description": ""}, "n": {"group": "Ungrouped variables", "templateType": "anything", "definition": "random(5..25)", "name": "n", "description": ""}, "m": {"group": "Ungrouped variables", "templateType": "anything", "definition": "random(0.5..3.5#0.1 except 1)", "name": "m", "description": ""}}, "ungrouped_variables": ["where", "m", "t", "tol", "n"], "question_groups": [{"pickingStrategy": "all-ordered", "questions": [], "name": "", "pickQuestions": 0}], "name": "Unbiased estimator for exponential distribution", "functions": {}, "showQuestionGroupNames": false, "parts": [{"scripts": {}, "gaps": [{"answer": "1/Y*(1-1/n)", "showCorrectAnswer": true, "vsetrange": [0, 1], "scripts": {}, "checkvariablenames": false, "expectedvariablenames": [], "showpreview": true, "checkingtype": "absdiff", "checkingaccuracy": 0.001, "type": "jme", "answersimplification": "std", "marks": 1, "vsetrangepoints": 5}], "type": "gapfill", "showCorrectAnswer": true, "prompt": "\n      <p>If $Y$ is the random variable given by the sample mean on $n$ inter-arrival times write down an expression for an unbiased estimator $T$ of $\\theta$.</p>\n        <p>$T=\\;\\;$[[0]]</p>\n      ", "marks": 0}, {"scripts": {}, "gaps": [{"showPrecisionHint": false, "allowFractions": false, "scripts": {}, "type": "numberentry", "showCorrectAnswer": true, "minValue": "{t-tol}", "correctAnswerFraction": false, "marks": 1, "maxValue": "{t+tol}"}], "type": "gapfill", "showCorrectAnswer": true, "prompt": "\n        \n        \n        <p>A random sample of $\\var{n}$ inter-arrival times gives a mean of $\\var{m}$ minutes.</p>\n        \n        \n        \n        <p>From this sample, find an estimate $t$ of the parameter $\\theta$.</p>\n        \n        \n        \n        <p>$t=\\;\\;$[[0]] (Enter to 2 decimal places).</p>\n        \n        \n      ", "marks": 0}], "variablesTest": {"condition": "", "maxRuns": 100}, "statement": "<p>The times between arrivals {where} can be described by an exponential distribution $X$ with parameter $\\theta$</p>", "tags": ["MAS1604", "arrivals", "checked2015", "cr1", "distributions", "estimate", "estimators", "exponential distribution", "inter arrival time", "inter-arrival times", "mean ", "random sample", "random variable", "random variables", "sample", "sample mean", "sc", "statistics", "tested1", "unbiased estimators"], "rulesets": {"std": ["all", "fractionNumbers", "!noLeadingMinus"]}, "preamble": {"css": "", "js": ""}, "type": "question", "metadata": {"notes": "<p><strong>13/07/2012:</strong></p>\n        <p>Added tags.</p>\n        <p>Improved display of correct answer.</p>\n        <p>Set new tolerance variable tol=0 for numeric input.</p>\n        <p>Checked calculation.</p>\n        <p>Description written.</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>Improved statement slightly. Also Advice - solution better displayed. Checked calculation. Added tested1 tag.</p>\n        <p>Rounding OK, added cr1 tag.</p>\n        <p>Perhaps could have scenarios? Added sc tag.</p>", "licence": "Creative Commons Attribution 4.0 International", "description": "<p>Arrivals given by exponential distribution, parameter $\\theta$ and $Y$, sample mean on inter-arrival times. Find and calculate unbiased estimator for $\\theta$.</p>"}, "advice": "<p>a)<br/>An unbiased estimator for the parameter $\\theta$ is:</p>\n    <p>\\[T =\\frac{1}{Y}&nbsp;\\left(1-\\frac{1}{n}\\right)\\]</p>\n    <p>b)<br/>Hence an estimate of $\\theta$ is given by:</p>\n    <p>\\[\\begin{eqnarray*} t&amp;=&amp;\\frac{1}{\\var{m}}\\left(1-\\frac{1}{\\var{n}}\\right)\\\\ &amp;=&amp;\\var{t} \\end{eqnarray*} \\] to 2 decimal places.</p>", "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/"}]}