// Numbas version: exam_results_page_options
{"name": "Poisson (sales)", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"name": "Poisson (sales)", "tags": ["expectation", "expected number", "poisson distribution", "Poisson distribution", "probabilities", "probability", "Probability", "rebel", "Rebel", "REBEL", "rebelmaths", "sc", "standard deviation", "statistical distributions", "statistics"], "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"}, "statement": "<p>{pre} $\\var{thismany}$.</p>\n<p>{something} $\\var{number1}$ {else}</p>\n<p>&nbsp;The Poisson distribution formula: $P(r)=\\frac{\\lambda^re^{-\\lambda}}{r!}$ or&nbsp;$P(r)=e^{-\\lambda}\\left[\\frac{\\lambda^r}{r!}\\right]$</p>", "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>", "rulesets": {}, "extensions": [], "builtin_constants": {"e": true, "pi,\u03c0": true, "i": true}, "constants": [], "variables": {"pre": {"name": "pre", "group": "Ungrouped variables", "definition": "\"The mean number of sales per day at a telecommunications centre is \"", "description": "", "templateType": "anything", "can_override": false}, "what": {"name": "what", "group": "Ungrouped variables", "definition": "\"daily sales.\"", "description": "", "templateType": "anything", "can_override": false}, "this": {"name": "this", "group": "Ungrouped variables", "definition": "\"a randomly selected employee makes exactly \"", "description": "", "templateType": "anything", "can_override": false}, "things": {"name": "things", "group": "Ungrouped variables", "definition": "\"sales.\"", "description": "", "templateType": "anything", "can_override": false}, "number1": {"name": "number1", "group": "Ungrouped variables", "definition": "if(thismany<8,2, 3)", "description": "", "templateType": "anything", "can_override": false}, "descx": {"name": "descx", "group": "Ungrouped variables", "definition": "\"the number of sales per day\"", "description": "", "templateType": "anything", "can_override": false}, "else": {"name": "else", "group": "Ungrouped variables", "definition": "\"per day.\"", "description": "", "templateType": "anything", "can_override": false}, "thismany": {"name": "thismany", "group": "Ungrouped variables", "definition": "random(5..10)", "description": "", "templateType": "anything", "can_override": false}, "something": {"name": "something", "group": "Ungrouped variables", "definition": "\"Employees receive a warning if they make less than \"", "description": "", "templateType": "anything", "can_override": false}, "tol": {"name": "tol", "group": "Ungrouped variables", "definition": "0.001", "description": "", "templateType": "anything", "can_override": false}, "v": {"name": "v", "group": "Ungrouped variables", "definition": "if(number1=2,0,1)", "description": "", "templateType": "anything", "can_override": false}, "tprob1": {"name": "tprob1", "group": "Ungrouped variables", "definition": "(thismany^thisnumber)*e^(-thismany)/fact(thisnumber)", "description": "", "templateType": "anything", "can_override": false}, "sd": {"name": "sd", "group": "Ungrouped variables", "definition": "precround(sqrt(thismany),3)", "description": "", "templateType": "anything", "can_override": false}, "tprob2": {"name": "tprob2", "group": "Ungrouped variables", "definition": "if(number1=2,e^(-thismany)*(1+thismany),e^(-thismany)*(1+thismany+thismany^2/2))", "description": "", "templateType": "anything", "can_override": false}, "prob2": {"name": "prob2", "group": "Ungrouped variables", "definition": "precround(tprob2,3)", "description": "", "templateType": "anything", "can_override": false}, "thisnumber": {"name": "thisnumber", "group": "Ungrouped variables", "definition": "if(thismany<8,thismany-1, random(3..7))", "description": "", "templateType": "anything", "can_override": false}, "thisaswell": {"name": "thisaswell", "group": "Ungrouped variables", "definition": "\"a randomly selected employee receives a warning.\"", "description": "", "templateType": "anything", "can_override": false}, "prob1": {"name": "prob1", "group": "Ungrouped variables", "definition": "precround(tprob1,3)", "description": "", "templateType": "anything", "can_override": false}}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": ["pre", "what", "this", "things", "number1", "descx", "else", "thismany", "something", "tol", "v", "tprob1", "sd", "tprob2", "prob2", "thisnumber", "thisaswell", "prob1"], "variable_groups": [], "functions": {}, "preamble": {"js": "", "css": ""}, "parts": [{"type": "gapfill", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<p>Assuming a Poisson distribution for {descX}, write down the value of $\\lambda$.</p>\n<p>$\\lambda = $[[0]]?</p>\n<p></p>", "gaps": [{"type": "numberentry", "useCustomName": false, "customName": "", "marks": "1", "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "thismany", "maxValue": "thismany", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "displayAnswer": "", "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}], "sortAnswers": false}, {"type": "gapfill", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<p>Find the probability that {this} $\\var{thisnumber}$ {things}</p>\n<p>$\\operatorname{P}(r=\\var{thisnumber})=$? [[0]] (to 3 decimal places).</p>\n<p>&nbsp;</p>\n<p>Find the probability that {thisaswell}</p>\n<p>$\\operatorname{P}(r &lt; \\var{number1})= $? [[1]] (to 3 decimal places).</p>", "gaps": [{"type": "numberentry", "useCustomName": false, "customName": "", "marks": "3", "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "prob1-tol", "maxValue": "prob1+tol", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "displayAnswer": "", "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}, {"type": "numberentry", "useCustomName": false, "customName": "", "marks": "5", "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "prob2-0.005", "maxValue": "prob2+0.005", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "displayAnswer": "", "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}], "sortAnswers": false}], "partsMode": "all", "maxMarks": 0, "objectives": [], "penalties": [], "objectiveVisibility": "always", "penaltyVisibility": "always", "contributors": [{"name": "Julie Crowley", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/113/"}, {"name": "Catherine Palmer", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/423/"}, {"name": "Patricia Cogan", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/3359/"}]}]}], "contributors": [{"name": "Julie Crowley", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/113/"}, {"name": "Catherine Palmer", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/423/"}, {"name": "Patricia Cogan", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/3359/"}]}