// Numbas version: exam_results_page_options
{"name": "Brad's copy of Poisson (sales)", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"parts": [{"customName": "", "variableReplacements": [], "extendBaseMarkingAlgorithm": true, "marks": 0, "sortAnswers": false, "unitTests": [], "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 = $[[0]]?</p>\n<p></p>", "gaps": [{"customName": "", "variableReplacements": [], "mustBeReduced": false, "extendBaseMarkingAlgorithm": true, "marks": "1", "correctAnswerStyle": "plain", "minValue": "thismany", "unitTests": [], "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "customMarkingAlgorithm": "", "scripts": {}, "showCorrectAnswer": true, "type": "numberentry", "variableReplacementStrategy": "originalfirst", "mustBeReducedPC": 0, "useCustomName": false, "correctAnswerFraction": false, "showFeedbackIcon": true, "allowFractions": false, "maxValue": "thismany"}], "customMarkingAlgorithm": "", "scripts": {}, "showCorrectAnswer": true, "type": "gapfill", "variableReplacementStrategy": "originalfirst", "useCustomName": false, "showFeedbackIcon": true}, {"customName": "", "variableReplacements": [], "extendBaseMarkingAlgorithm": true, "marks": 0, "sortAnswers": false, "unitTests": [], "prompt": "\n        <p>Find the probability that {this} $\\var{thisnumber}$ {things}</p>\n      <p>$\\operatorname{P}(X=\\var{thisnumber})=$? [[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>\n        ", "gaps": [{"customName": "", "variableReplacements": [], "mustBeReduced": false, "extendBaseMarkingAlgorithm": true, "marks": 1, "correctAnswerStyle": "plain", "minValue": "prob1-tol", "unitTests": [], "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "customMarkingAlgorithm": "", "scripts": {}, "showCorrectAnswer": true, "type": "numberentry", "variableReplacementStrategy": "originalfirst", "mustBeReducedPC": 0, "useCustomName": false, "correctAnswerFraction": false, "showFeedbackIcon": true, "allowFractions": false, "maxValue": "prob1+tol"}, {"customName": "", "variableReplacements": [], "mustBeReduced": false, "extendBaseMarkingAlgorithm": true, "marks": 1, "correctAnswerStyle": "plain", "minValue": "prob2-tol", "unitTests": [], "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "customMarkingAlgorithm": "", "scripts": {}, "showCorrectAnswer": true, "type": "numberentry", "variableReplacementStrategy": "originalfirst", "mustBeReducedPC": 0, "useCustomName": false, "correctAnswerFraction": false, "showFeedbackIcon": true, "allowFractions": false, "maxValue": "prob2+tol"}], "customMarkingAlgorithm": "", "scripts": {}, "showCorrectAnswer": true, "type": "gapfill", "variableReplacementStrategy": "originalfirst", "useCustomName": false, "showFeedbackIcon": true}], "functions": {}, "preamble": {"css": "", "js": ""}, "variables": {"else": {"templateType": "anything", "description": "", "name": "else", "group": "Ungrouped variables", "definition": "\"per day.\""}, "something": {"templateType": "anything", "description": "", "name": "something", "group": "Ungrouped variables", "definition": "\"Employees receive a warning if they make less than \""}, "sd": {"templateType": "anything", "description": "", "name": "sd", "group": "Ungrouped variables", "definition": "precround(sqrt(thismany),3)"}, "prob2": {"templateType": "anything", "description": "", "name": "prob2", "group": "Ungrouped variables", "definition": "precround(tprob2,3)"}, "descx": {"templateType": "anything", "description": "", "name": "descx", "group": "Ungrouped variables", "definition": "\"the number of sales per day\""}, "pre": {"templateType": "anything", "description": "", "name": "pre", "group": "Ungrouped variables", "definition": "\"The mean number of sales per day at a telecommunications centre is \""}, "v": {"templateType": "anything", "description": "", "name": "v", "group": "Ungrouped variables", "definition": "if(number1=2,0,1)"}, "prob1": {"templateType": "anything", "description": "", "name": "prob1", "group": "Ungrouped variables", "definition": "precround(tprob1,3)"}, "number1": {"templateType": "anything", "description": "", "name": "number1", "group": "Ungrouped variables", "definition": "if(thismany<8,2, 3)"}, "this": {"templateType": "anything", "description": "", "name": "this", "group": "Ungrouped variables", "definition": "\"a randomly selected employee makes exactly \""}, "things": {"templateType": "anything", "description": "", "name": "things", "group": "Ungrouped variables", "definition": "\"sales.\""}, "tprob1": {"templateType": "anything", "description": "", "name": "tprob1", "group": "Ungrouped variables", "definition": "(thismany^thisnumber)*e^(-thismany)/fact(thisnumber)"}, "tprob2": {"templateType": "anything", "description": "", "name": "tprob2", "group": "Ungrouped variables", "definition": "if(number1=2,e^(-thismany)*(1+thismany),e^(-thismany)*(1+thismany+thismany^2/2))"}, "what": {"templateType": "anything", "description": "", "name": "what", "group": "Ungrouped variables", "definition": "\"daily sales.\""}, "tol": {"templateType": "anything", "description": "", "name": "tol", "group": "Ungrouped variables", "definition": "0.001"}, "thismany": {"templateType": "anything", "description": "", "name": "thismany", "group": "Ungrouped variables", "definition": "random(5..10)"}, "thisaswell": {"templateType": "anything", "description": "", "name": "thisaswell", "group": "Ungrouped variables", "definition": "\"a randomly selected employee receives a warning.\""}, "thisnumber": {"templateType": "anything", "description": "", "name": "thisnumber", "group": "Ungrouped variables", "definition": "if(thismany<8,thismany-1, random(3..7))"}}, "statement": "\n    <p>{pre} $\\var{thismany}$.</p>\n  <p>{something} $\\var{number1}$ {else}</p>\n  <p>&nbsp;</p>\n    ", "variable_groups": [], "extensions": [], "name": "Brad's copy of Poisson (sales)", "metadata": {"description": "<p>Application of the Poisson distribution given expected number of events per interval.</p>\n<p>Finding probabilities using the Poisson distribution.</p>\n<p>rebelmaths</p>", "licence": "Creative Commons Attribution 4.0 International"}, "ungrouped_variables": ["pre", "what", "this", "things", "number1", "descx", "else", "thismany", "something", "tol", "v", "tprob1", "sd", "tprob2", "prob2", "thisnumber", "thisaswell", "prob1"], "rulesets": {}, "variablesTest": {"maxRuns": 100, "condition": ""}, "advice": "<p>a)</p>\n<p>1. $X \\sim \\operatorname{Poisson}(\\var{thismany})$, so $\\lambda = \\var{thismany}$.</p>\n<p>b)</p>\n<p>Remember that for a Poisson random variable:<br/>\\begin{align}<br/>\\operatorname{P}(X=x)&amp;=\\dfrac{\\lambda^x\\times e^{-\\lambda}}{x!}\\\\<br/>\\end{align}</p>\n<p>1.\\[ \\begin{eqnarray*}\\operatorname{P}(X = \\var{thisnumber}) &amp;=&amp; \\frac{\\var{thismany} ^ {\\var{thisnumber}}e ^ { -\\var{thismany}}} {\\var{thisnumber}!}\\\\&amp; =&amp; \\var{prob1} \\end{eqnarray*} \\] to 3 decimal places.</p>\n<p>&nbsp;</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>", "tags": [], "contributors": [{"name": "Julie Crowley", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/113/"}, {"name": "Brad Allison", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/3394/"}]}]}], "contributors": [{"name": "Julie Crowley", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/113/"}, {"name": "Brad Allison", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/3394/"}]}