// Numbas version: exam_results_page_options
{"name": "Alex'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": [{"variablesTest": {"condition": "", "maxRuns": 100}, "extensions": [], "advice": "<p>a)</p>\n<p>1. $X \\sim \\operatorname{Poisson}(\\var{thismany})$, so $\\lambda = \\var{thismany}$.</p>\n<p></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>", "statement": "\n    <p>{pre} $\\var{thismany}$.</p>\n  <p>{something} $\\var{number1}$ {else}</p>\n  <p>&nbsp;</p>\n    ", "question_groups": [{"pickQuestions": 0, "pickingStrategy": "all-ordered", "questions": [], "name": ""}], "variables": {"what": {"group": "Ungrouped variables", "templateType": "anything", "description": "", "definition": "\"daily sales.\"", "name": "what"}, "this": {"group": "Ungrouped variables", "templateType": "anything", "description": "", "definition": "\"a randomly selected employee makes exactly \"", "name": "this"}, "v": {"group": "Ungrouped variables", "templateType": "anything", "description": "", "definition": "if(number1=2,0,1)", "name": "v"}, "pre": {"group": "Ungrouped variables", "templateType": "anything", "description": "", "definition": "\"The mean number of sales per day at a telecommunications centre is \"", "name": "pre"}, "tprob1": {"group": "Ungrouped variables", "templateType": "anything", "description": "", "definition": "(thismany^thisnumber)*e^(-thismany)/fact(thisnumber)", "name": "tprob1"}, "tprob2": {"group": "Ungrouped variables", "templateType": "anything", "description": "", "definition": "if(number1=2,e^(-thismany)*(1+thismany),e^(-thismany)*(1+thismany+thismany^2/2))", "name": "tprob2"}, "number1": {"group": "Ungrouped variables", "templateType": "anything", "description": "", "definition": "if(thismany<8,2, 3)", "name": "number1"}, "thismany": {"group": "Ungrouped variables", "templateType": "anything", "description": "", "definition": "random(5..10)", "name": "thismany"}, "else": {"group": "Ungrouped variables", "templateType": "anything", "description": "", "definition": "\"per day.\"", "name": "else"}, "descx": {"group": "Ungrouped variables", "templateType": "anything", "description": "", "definition": "\"the number of sales per day\"", "name": "descx"}, "tol": {"group": "Ungrouped variables", "templateType": "anything", "description": "", "definition": "0.001", "name": "tol"}, "thisaswell": {"group": "Ungrouped variables", "templateType": "anything", "description": "", "definition": "\"a randomly selected employee receives a warning.\"", "name": "thisaswell"}, "things": {"group": "Ungrouped variables", "templateType": "anything", "description": "", "definition": "\"sales.\"", "name": "things"}, "prob1": {"group": "Ungrouped variables", "templateType": "anything", "description": "", "definition": "precround(tprob1,3)", "name": "prob1"}, "sd": {"group": "Ungrouped variables", "templateType": "anything", "description": "", "definition": "precround(sqrt(thismany),3)", "name": "sd"}, "thisnumber": {"group": "Ungrouped variables", "templateType": "anything", "description": "", "definition": "if(thismany<8,thismany-1, random(3..7))", "name": "thisnumber"}, "something": {"group": "Ungrouped variables", "templateType": "anything", "description": "", "definition": "\"Employees receive a warning if they make less than \"", "name": "something"}, "prob2": {"group": "Ungrouped variables", "templateType": "anything", "description": "", "definition": "precround(tprob2,3)", "name": "prob2"}}, "name": "Alex's copy of Poisson (sales)", "rulesets": {}, "type": "question", "variable_groups": [], "parts": [{"marks": 0, "variableReplacementStrategy": "originalfirst", "type": "gapfill", "showCorrectAnswer": true, "variableReplacements": [], "gaps": [{"marks": "1", "maxValue": "thismany", "type": "numberentry", "showCorrectAnswer": true, "variableReplacements": [], "minValue": "thismany", "correctAnswerFraction": false, "allowFractions": false, "showPrecisionHint": false, "scripts": {}, "variableReplacementStrategy": "originalfirst"}], "scripts": {}, "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>"}, {"marks": 0, "variableReplacementStrategy": "originalfirst", "type": "gapfill", "showCorrectAnswer": true, "variableReplacements": [], "gaps": [{"marks": 1, "maxValue": "prob1+tol", "type": "numberentry", "showCorrectAnswer": true, "variableReplacements": [], "minValue": "prob1-tol", "correctAnswerFraction": false, "allowFractions": false, "showPrecisionHint": false, "scripts": {}, "variableReplacementStrategy": "originalfirst"}, {"marks": 1, "maxValue": "prob2+tol", "type": "numberentry", "showCorrectAnswer": true, "variableReplacements": [], "minValue": "prob2-tol", "correctAnswerFraction": false, "allowFractions": false, "showPrecisionHint": false, "scripts": {}, "variableReplacementStrategy": "originalfirst"}], "scripts": {}, "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        "}], "tags": ["expectation", "expected number", "Poisson distribution", "poisson distribution", "probabilities", "probability", "Probability", "Rebel", "rebel", "REBEL", "rebelmaths", "sc", "standard deviation", "statistical distributions", "statistics"], "preamble": {"js": "", "css": ""}, "ungrouped_variables": ["pre", "what", "this", "things", "number1", "descx", "else", "thismany", "something", "tol", "v", "tprob1", "sd", "tprob2", "prob2", "thisnumber", "thisaswell", "prob1"], "functions": {}, "showQuestionGroupNames": false, "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"}, "contributors": [{"name": "Alex Van den Hof", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/1273/"}]}]}], "contributors": [{"name": "Alex Van den Hof", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/1273/"}]}