// Numbas version: exam_results_page_options
{"name": "Probability, expectation and standard deviation of Poisson distribution", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"variablesTest": {"condition": "", "maxRuns": 100}, "tags": ["checked2015", "MAS1403"], "variables": {"tprob2": {"description": "", "templateType": "anything", "group": "Ungrouped variables", "definition": "if(number1=2,e^(-thismany)*(1+thismany),e^(-thismany)*(1+thismany+thismany^2/2))", "name": "tprob2"}, "pre": {"description": "", "templateType": "anything", "group": "Ungrouped variables", "definition": "\"The mean number of sales per day at a telecommunications centre is \"", "name": "pre"}, "thismany": {"description": "", "templateType": "anything", "group": "Ungrouped variables", "definition": "random(5..10)", "name": "thismany"}, "else": {"description": "", "templateType": "anything", "group": "Ungrouped variables", "definition": "\"per day.\"", "name": "else"}, "prob1": {"description": "", "templateType": "anything", "group": "Ungrouped variables", "definition": "precround(tprob1,3)", "name": "prob1"}, "descx": {"description": "", "templateType": "anything", "group": "Ungrouped variables", "definition": "\"the number of sales per day\"", "name": "descx"}, "tprob1": {"description": "", "templateType": "anything", "group": "Ungrouped variables", "definition": "(thismany^thisnumber)*e^(-thismany)/fact(thisnumber)", "name": "tprob1"}, "v": {"description": "", "templateType": "anything", "group": "Ungrouped variables", "definition": "if(number1=2,0,1)", "name": "v"}, "prob2": {"description": "", "templateType": "anything", "group": "Ungrouped variables", "definition": "precround(tprob2,3)", "name": "prob2"}, "what": {"description": "", "templateType": "anything", "group": "Ungrouped variables", "definition": "\"daily sales.\"", "name": "what"}, "thisnumber": {"description": "", "templateType": "anything", "group": "Ungrouped variables", "definition": "if(thismany<8,thismany-1, random(3..7))", "name": "thisnumber"}, "something": {"description": "", "templateType": "anything", "group": "Ungrouped variables", "definition": "\"Employees receive a warning if they make less than \"", "name": "something"}, "number1": {"description": "", "templateType": "anything", "group": "Ungrouped variables", "definition": "if(thismany<8,2, 3)", "name": "number1"}, "sd": {"description": "", "templateType": "anything", "group": "Ungrouped variables", "definition": "precround(sqrt(thismany),3)", "name": "sd"}, "thisaswell": {"description": "", "templateType": "anything", "group": "Ungrouped variables", "definition": "\"a randomly selected employee receives a warning.\"", "name": "thisaswell"}, "things": {"description": "", "templateType": "anything", "group": "Ungrouped variables", "definition": "\"sales.\"", "name": "things"}, "this": {"description": "", "templateType": "anything", "group": "Ungrouped variables", "definition": "\"a randomly selected employee makes exactly \"", "name": "this"}, "tol": {"description": "", "templateType": "anything", "group": "Ungrouped variables", "definition": "0.001", "name": "tol"}}, "name": "Probability, expectation and standard deviation of Poisson distribution", "advice": "\n  <p>a)</p>\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    <p>b)</p>\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>&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>\n  ", "metadata": {"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", "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"}, "statement": "\n  <p>{pre} $\\var{thismany}$.</p>\n    <p>{something} $\\var{number1}$ {else}</p>\n    <p>&nbsp;</p>\n  ", "variable_groups": [], "showQuestionGroupNames": false, "functions": {}, "ungrouped_variables": ["pre", "what", "this", "things", "number1", "descx", "else", "thismany", "something", "tol", "v", "tprob1", "sd", "tprob2", "prob2", "thisnumber", "thisaswell", "prob1"], "type": "question", "preamble": {"js": "", "css": ""}, "rulesets": {}, "parts": [{"gaps": [{"allowFractions": false, "showPrecisionHint": false, "type": "numberentry", "correctAnswerFraction": false, "marks": 0.25, "minValue": "thismany", "maxValue": "thismany", "scripts": {}, "showCorrectAnswer": true}, {"allowFractions": false, "showPrecisionHint": false, "type": "numberentry", "correctAnswerFraction": false, "marks": 0.25, "minValue": "thismany", "maxValue": "thismany", "scripts": {}, "showCorrectAnswer": true}, {"allowFractions": false, "showPrecisionHint": false, "type": "numberentry", "correctAnswerFraction": false, "marks": 0.5, "minValue": "sd-tol", "maxValue": "sd+tol", "scripts": {}, "showCorrectAnswer": true}], "showCorrectAnswer": true, "type": "gapfill", "marks": 0, "scripts": {}, "prompt": "\n      <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>Find $\\operatorname{E}[X]$ the expected {descX}.</p>\n        <p>$\\operatorname{E}[X]=$?[[1]]</p>\n        <p>Find the standard deviation for {what}.</p>\n        <p>Standard deviation = ? [[2]] (to 3 decimal places).</p>\n      "}, {"gaps": [{"allowFractions": false, "showPrecisionHint": false, "type": "numberentry", "correctAnswerFraction": false, "marks": 1, "minValue": "prob1-tol", "maxValue": "prob1+tol", "scripts": {}, "showCorrectAnswer": true}, {"allowFractions": false, "showPrecisionHint": false, "type": "numberentry", "correctAnswerFraction": false, "marks": 1, "minValue": "prob2-tol", "maxValue": "prob2+tol", "scripts": {}, "showCorrectAnswer": true}], "showCorrectAnswer": true, "type": "gapfill", "marks": 0, "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      "}], "question_groups": [{"pickingStrategy": "all-ordered", "questions": [], "pickQuestions": 0, "name": ""}], "contributors": [{"name": "Newcastle University Mathematics and Statistics", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/697/"}, {"name": "Simon Thomas", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/3148/"}]}]}], "contributors": [{"name": "Newcastle University Mathematics and Statistics", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/697/"}, {"name": "Simon Thomas", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/3148/"}]}