// Numbas version: exam_results_page_options
{"name": "Calculate probability, expected value and variance of a geometric 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": {"tol": {"templateType": "anything", "group": "Ungrouped variables", "definition": "0.001", "description": "", "name": "tol"}, "ans": {"templateType": "anything", "group": "Ungrouped variables", "definition": "precround(tans,3)", "description": "", "name": "ans"}, "v": {"templateType": "anything", "group": "Ungrouped variables", "definition": "arz(y2,y3,[1,2,3,4,5,6,7])", "description": "", "name": "v"}, "y3": {"templateType": "anything", "group": "Ungrouped variables", "definition": "y2+u", "description": "", "name": "y3"}, "y2": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(1..3)", "description": "", "name": "y2"}, "p": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(0.3..0.7#0.05)", "description": "", "name": "p"}, "tans": {"templateType": "anything", "group": "Ungrouped variables", "definition": "precround(geometricPDF(y1,p),4)", "description": "", "name": "tans"}, "y1": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(1..7)", "description": "", "name": "y1"}, "u": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(1..4)", "description": "", "name": "u"}, "ans1": {"templateType": "anything", "group": "Ungrouped variables", "definition": "precround(tans1,3)", "description": "", "name": "ans1"}, "tans1": {"templateType": "anything", "group": "Ungrouped variables", "definition": "precround(geometricCDF(y3,p)-geometricCDF(y2-1,p),4)", "description": "", "name": "tans1"}}, "ungrouped_variables": ["ans1", "ans", "p", "u", "tol", "v", "y1", "tans", "y3", "y2", "tans1"], "question_groups": [{"pickingStrategy": "all-ordered", "questions": [], "name": "", "pickQuestions": 0}], "name": "Calculate probability, expected value and variance of a geometric distribution, ", "functions": {"arz": {"type": "list", "language": "jme", "definition": "map(if(x<m or x >n,0,1),x,a)", "parameters": [["m", "number"], ["n", "number"], ["a", "list"]]}}, "showQuestionGroupNames": false, "parts": [{"scripts": {}, "gaps": [{"precisionPartialCredit": 0, "allowFractions": false, "correctAnswerFraction": false, "minValue": "ans", "maxValue": "ans", "precision": "3", "type": "numberentry", "precisionType": "dp", "showPrecisionHint": false, "strictPrecision": false, "scripts": {}, "precisionMessage": "You have not given your answer to the correct precision.", "showCorrectAnswer": true, "marks": 1}], "type": "gapfill", "prompt": "<p>Compute $\\operatorname{P}(W=\\var{y1}) = $&nbsp;[[0]]</p>", "showCorrectAnswer": true, "marks": 0}, {"scripts": {}, "gaps": [{"precisionPartialCredit": 0, "allowFractions": false, "correctAnswerFraction": false, "minValue": "ans1", "maxValue": "ans1", "precision": "3", "type": "numberentry", "precisionType": "dp", "showPrecisionHint": false, "strictPrecision": false, "scripts": {}, "precisionMessage": "You have not given your answer to the correct precision.", "showCorrectAnswer": true, "marks": 1}], "type": "gapfill", "prompt": "<p>Compute $\\operatorname{P}(\\var{y2} \\le W \\le \\var{y3}) =$&nbsp;[[0]]<span> (enter your answer to 3 decimal places)</span></p>", "showCorrectAnswer": true, "marks": 0}, {"scripts": {}, "gaps": [{"precisionPartialCredit": 0, "allowFractions": false, "correctAnswerFraction": false, "minValue": "1/p", "maxValue": "1/p", "precision": "3", "type": "numberentry", "precisionType": "dp", "showPrecisionHint": false, "strictPrecision": false, "scripts": {}, "precisionMessage": "You have not given your answer to the correct precision.", "showCorrectAnswer": true, "marks": 1}, {"precisionPartialCredit": 0, "allowFractions": false, "correctAnswerFraction": false, "minValue": "(1-p)/p^2", "maxValue": "(1-p)/p^2", "precision": "3", "type": "numberentry", "precisionType": "dp", "showPrecisionHint": false, "strictPrecision": false, "scripts": {}, "precisionMessage": "You have not given your answer to the correct precision.", "showCorrectAnswer": true, "marks": 1}], "type": "gapfill", "prompt": "<p>Find:</p>\n<p>1. $\\operatorname{E}[W] = $&nbsp;[[0]]</p>\n<p>2. $\\operatorname{Var}(W)=$&nbsp;[[1]]</p>", "showCorrectAnswer": true, "marks": 0}], "statement": "<p>Enter your answers to the following questions to 3 decimal places.</p>\n<p>Suppose $W\\sim \\operatorname{Geometric}(\\var{p})$</p>", "tags": ["checked2015", "discrete random variables", "distributions", "expectation of geometric distribution", "geometric distribution", "Geometric Distribution", "MAS1604", "MAS2304", "probability", "Probability", "random variables", "sr", "statistics", "tested1", "udf", "variance of geometric distribution"], "rulesets": {"std": ["all", "fractionNumbers", "!collectNumbers", "!noLeadingMinus"]}, "preamble": {"css": "", "js": ""}, "type": "question", "metadata": {"notes": "<p><strong>7/07/2012:</strong></p>\n        <p>Added tags.</p>\n        <p>Cannot access stats extension at present, so question does not run. Issue posted.</p>\n        <p>Set new tolerance variable tol=0.001 for first two answers.</p>\n        <p>Calculation to be tested under Test Run.</p>\n        <p><strong>22/07/2012:</strong></p>\n        <p>Added description.</p>\n        <p>Can check calculation now stats extension box ticked.</p>\n        <p>Calculations checked.</p>\n        <p><strong>31/07/2012:</strong></p>\n        <p>&nbsp;Added tags.</p>\n        <p>Question appears to be working correctly.</p>\n        <p><strong>20/12/2012:</strong></p>\n        <p>Checked calculations against standard tables, OK.</p>\n        <p>Added tested1 tag.</p>\n        <p>User defined function arz picks out values in an array a lying between given values m and n.</p>\n        <p>arz(m,n,a)=map(if(x&lt;m or x &gt;n,0,1),x,a). Added udf tag.</p>\n        <p>Extra rounding introduced for variables tans1, tans. Added tag sr.</p>", "licence": "Creative Commons Attribution 4.0 International", "description": "<p>$W \\sim \\operatorname{Geometric}(p)$. Find $P(W=a)$, $P(b \\le W \\le c)$, $E[W]$, $\\operatorname{Var}(W)$.</p>"}, "variablesTest": {"condition": "", "maxRuns": 100}, "advice": "<h4>a)</h4>\n<p>If $W \\sim \\operatorname{Geometric}(p)$ then</p>\n<p>\\[\\operatorname{P}(W=w)=p \\times (1-p)^{w-1}\\]</p>\n<p>Hence:</p>\n<p>\\[ \\operatorname{P}(W= \\var{y1}) = \\simplify[!otherNumbers,zeroPower,unitFactor]{{p}*(1-{p})^{y1-1}} = \\var{tans} \\approx \\var{ans}\\]</p>\n<p>to 3 decimal places.</p>\n<h4>b)</h4>\n<p>We have:</p>\n<p>\\begin{align}<br/>\\operatorname{P}(\\var{y2} \\le W \\le \\var{y3}) &amp;= \\sum_{w=\\var{y2}}^{w=\\var{y3}}\\var{p}\\times(1-\\var{p})^{w-1} \\\\ <br/>&amp;= \\simplify[unitFactor,zeroTerm,zeroFactor]{{v[0]}*{p}+{v[1]}*{p}*{1-p} +{v[2]}*{p}*{1-p}^2 +{v[3]}*{p}*{1-p}^3 +{v[4]}*{p}*{1-p}^4 +{v[5]}*{p}*{1-p}^5+{v[6]}*{p}*{1-p}^6} \\\\ <br/>&amp;= \\var{tans1} \\\\ <br/>&amp;\\approx \\var{ans1}<br/>\\end{align}</p>\n<p>to 3 decimal places.</p>\n<h4>c)</h4>\n<p>For the Geometric distribution $\\operatorname{Geometric}(p)$ we have:</p>\n<p>\\begin{align}<br/>\\operatorname{E}[W] &amp;= \\frac{1}{p} \\\\ \\\\<br/> \\operatorname{Var}(W) &amp;= \\frac{1-p}{p^2} <br/>\\end{align}</p>\n<p>Hence in this case:</p>\n<p>\\begin{align}<br/>\\operatorname{E}[W] &amp;= \\simplify{1/{p}} =&nbsp;\\var{precround(1/p,3)} \\\\[0.5em]<br/>\\operatorname{Var}(W) &amp;= \\simplify[all,!collectNumbers,!otherNumbers]{(1-{p})/({p}^2)} =&nbsp;\\var{precround((1-p)/p^2,3)}<br/>\\end{align}</p>\n<p>to 3 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/"}]}