// Numbas version: exam_results_page_options
{"name": "BS3.2", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"functions": {}, "ungrouped_variables": ["pre", "what", "this", "things", "number1", "descx", "else", "thismany", "something", "tol", "v", "tprob1", "sd", "tprob2", "prob2", "thisnumber", "thisaswell", "prob1"], "name": "BS3.2", "tags": ["Poisson distribution", "Probability", "expectation", "expected number", "poisson distribution", "probabilities", "probability", "sc", "standard deviation", "statistical distributions", "statistics"], "preamble": {"css": "", "js": ""}, "advice": "<h4>a)</h4>\n<p>1. $X \\sim \\operatorname{Poisson}(\\var{thismany})$, so $\\lambda = \\var{thismany}$.</p>\n<p>2. The expectation is given by $\\operatorname{E}[X]=\\lambda=\\var{thismany}$</p>\n<p>3. $\\operatorname{stdev}(X)=\\sqrt{\\lambda}=\\sqrt{\\var{thismany}}=\\var{sd}$ to 3 decimal places.</p>\n<h4>b)</h4>\n<p>1. \\[ \\begin{eqnarray*}\\operatorname{P}(X = \\var{thisnumber}) &amp;=&amp; \\frac{e ^ { -\\var{thismany}}\\var{thismany} ^ {\\var{thisnumber}}} {\\var{thisnumber}!}\\\\&amp; =&amp; \\var{prob1} \\end{eqnarray*} \\] to 3 decimal places.</p>\n<p>2. If an employee receives a warning then he or she must have sold less than {number1}.</p>\n<p>Hence we need to find:</p>\n<p>\\[ \\begin{eqnarray*}\\operatorname{P}(X &lt; \\var{number1})&amp; =&amp; \\simplify[all,!collectNumbers]{P(X = 0) + P(X = 1) + {v}*P(X = 2)}\\\\&amp; =&amp; \\simplify[all,!collectNumbers]{e ^ { -thismany} + {thismany} * e ^ { -thismany} + {v} * (({thismany} ^ 2 * e ^ { -thismany}) / 2)} \\\\&amp;=&amp; \\var{prob2} \\end{eqnarray*} \\]</p>\n<p>to 3 decimal places.</p>\n<p>&nbsp;</p>", "rulesets": {}, "parts": [{"prompt": "<p>Assuming a Poisson distribution for $X$, {descX}, write down the value of $\\lambda$.</p>\n<p>$X \\sim \\operatorname{Poisson}(\\lambda)$</p>\n<p>$\\lambda = $&nbsp;[[0]]</p>\n<p>Find $\\operatorname{E}[X]$ the expected {descX}.</p>\n<p>$\\operatorname{E}[X]=$&nbsp;[[1]]</p>\n<p>Find the standard deviation for {what}.</p>\n<p>Standard deviation = [[2]] (to 3 decimal places).</p>", "marks": 0, "gaps": [{"allowFractions": false, "marks": 0.25, "maxValue": "thismany", "minValue": "thismany", "correctAnswerFraction": false, "showCorrectAnswer": true, "scripts": {}, "type": "numberentry", "showPrecisionHint": false}, {"allowFractions": false, "marks": 0.25, "maxValue": "thismany", "minValue": "thismany", "correctAnswerFraction": false, "showCorrectAnswer": true, "scripts": {}, "type": "numberentry", "showPrecisionHint": false}, {"allowFractions": false, "marks": 0.5, "maxValue": "sd+tol", "minValue": "sd-tol", "correctAnswerFraction": false, "showCorrectAnswer": true, "scripts": {}, "type": "numberentry", "showPrecisionHint": false}], "showCorrectAnswer": true, "scripts": {}, "type": "gapfill"}, {"prompt": "<p>Find the probability that {this} $\\var{thisnumber}$ {things}</p>\n<p>$\\operatorname{P}(X=\\var{thisnumber})=$&nbsp;[[0]] (to 3 decimal places).</p>\n<p>&nbsp;</p>\n<p>Find the probability that {thisaswell}</p>\n<p>Probability = [[1]] (to 3 decimal places).</p>", "marks": 0, "gaps": [{"allowFractions": false, "marks": 1, "maxValue": "prob1+tol", "minValue": "prob1-tol", "correctAnswerFraction": false, "showCorrectAnswer": true, "scripts": {}, "type": "numberentry", "showPrecisionHint": false}, {"allowFractions": false, "marks": 1, "maxValue": "prob2+tol", "minValue": "prob2-tol", "correctAnswerFraction": false, "showCorrectAnswer": true, "scripts": {}, "type": "numberentry", "showPrecisionHint": false}], "showCorrectAnswer": true, "scripts": {}, "type": "gapfill"}], "statement": "\n    <p>{pre} $\\var{thismany}$.</p>\n  <p>{something} $\\var{number1}$ {else}</p>\n  <p>&nbsp;</p>\n    ", "variable_groups": [], "variablesTest": {"maxRuns": 100, "condition": ""}, "variables": {"pre": {"definition": "\"The mean number of sales per day at a telecommunications centre is \"", "templateType": "anything", "group": "Ungrouped variables", "name": "pre", "description": ""}, "what": {"definition": "\"daily sales.\"", "templateType": "anything", "group": "Ungrouped variables", "name": "what", "description": ""}, "this": {"definition": "\"a randomly selected employee makes exactly \"", "templateType": "anything", "group": "Ungrouped variables", "name": "this", "description": ""}, "things": {"definition": "\"sales.\"", "templateType": "anything", "group": "Ungrouped variables", "name": "things", "description": ""}, "prob1": {"definition": "precround(tprob1,3)", "templateType": "anything", "group": "Ungrouped variables", "name": "prob1", "description": ""}, "v": {"definition": "if(number1=2,0,1)", "templateType": "anything", "group": "Ungrouped variables", "name": "v", "description": ""}, "descx": {"definition": "\"the number of sales per day\"", "templateType": "anything", "group": "Ungrouped variables", "name": "descx", "description": ""}, "else": {"definition": "\"per day.\"", "templateType": "anything", "group": "Ungrouped variables", "name": "else", "description": ""}, "thismany": {"definition": "random(5..10)", "templateType": "anything", "group": "Ungrouped variables", "name": "thismany", "description": ""}, "something": {"definition": "\"Employees receive a warning if they make less than \"", "templateType": "anything", "group": "Ungrouped variables", "name": "something", "description": ""}, "tol": {"definition": "0.001", "templateType": "anything", "group": "Ungrouped variables", "name": "tol", "description": ""}, "number1": {"definition": "if(thismany<8,2, 3)", "templateType": "anything", "group": "Ungrouped variables", "name": "number1", "description": ""}, "tprob1": {"definition": "(thismany^thisnumber)*e^(-thismany)/fact(thisnumber)", "templateType": "anything", "group": "Ungrouped variables", "name": "tprob1", "description": ""}, "tprob2": {"definition": "if(number1=2,e^(-thismany)*(1+thismany),e^(-thismany)*(1+thismany+thismany^2/2))", "templateType": "anything", "group": "Ungrouped variables", "name": "tprob2", "description": ""}, "prob2": {"definition": "precround(tprob2,3)", "templateType": "anything", "group": "Ungrouped variables", "name": "prob2", "description": ""}, "thisnumber": {"definition": "if(thismany<8,thismany-1, random(3..7))", "templateType": "anything", "group": "Ungrouped variables", "name": "thisnumber", "description": ""}, "thisaswell": {"definition": "\"a randomly selected employee receives a warning.\"", "templateType": "anything", "group": "Ungrouped variables", "name": "thisaswell", "description": ""}, "sd": {"definition": "precround(sqrt(thismany),3)", "templateType": "anything", "group": "Ungrouped variables", "name": "sd", "description": ""}}, "metadata": {"notes": "\n        \t\t<p><strong>31/12/2012:</strong></p>\n        \t\t    <p>Can be configured to other applications using the string variables supplied. Hence added tag sc.</p>\n        \t\t    <p>Not as yet properly tested.</p>\n        \t\t", "description": "\n        \t\t<p>Application of the Poisson distribution given expected number of events per interval.</p>\n        \t\t    <p>Finding probabilities using the Poisson distribution.</p>\n        \t\t", "licence": "Creative Commons Attribution 4.0 International"}, "type": "question", "showQuestionGroupNames": false, "question_groups": [{"name": "", "pickingStrategy": "all-ordered", "pickQuestions": 0, "questions": []}], "contributors": [{"name": "Bill Foster", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/6/"}]}]}], "contributors": [{"name": "Bill Foster", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/6/"}]}