// 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}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"variablesTest": {"condition": "", "maxRuns": 100}, "extensions": [], "advice": "

a)

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1. $X \\sim \\operatorname{Poisson}(\\var{thismany})$, so $\\lambda = \\var{thismany}$.

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b)

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Remember that for a Poisson random variable:
\\begin{align}
\\operatorname{P}(X=x)&=\\dfrac{\\lambda^x\\times e^{-\\lambda}}{x!}\\\\
\\end{align}

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1.\$\\begin{eqnarray*}\\operatorname{P}(X = \\var{thisnumber}) &=& \\frac{\\var{thismany} ^ {\\var{thisnumber}}e ^ { -\\var{thismany}}} {\\var{thisnumber}!}\\\\& =& \\var{prob1} \\end{eqnarray*} \$ to 3 decimal places.

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2. If an employee receives a warning then he or she must have sold less than {number1}.

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Hence we need to find :

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\$\\begin{eqnarray*}\\operatorname{P}(X < \\var{number1})& =& \\simplify[all,!collectNumbers]{P(X = 0) + P(X = 1) + {v}*P(X = 2)}\\\\& =& \\simplify[all,!collectNumbers]{e ^ { -thismany} + {thismany} * e ^ { -thismany} + {v} * (({thismany} ^ 2 * e ^ { -thismany}) / 2)} \\\\&=& \\var{prob2} \\end{eqnarray*} \$

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to 3 decimal places.

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", "statement": "\n

{pre} $\\var{thismany}$.

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{something} $\\var{number1}$ {else}

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\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": "

Assuming a Poisson distribution for $X$, {descX}, write down the value of $\\lambda$.

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$X \\sim \\operatorname{Poisson}(\\lambda)$

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$\\lambda =$[[0]]?

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"}, {"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

Find the probability that {this} $\\var{thisnumber}$ {things}

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$\\operatorname{P}(X=\\var{thisnumber})=$? [[0]] (to 3 decimal places).

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Find the probability that {thisaswell}

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Probability = ? [[1]] (to 3 decimal places).

\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": "

Application of the Poisson distribution given expected number of events per interval.

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Finding probabilities using the Poisson distribution.

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rebelmaths

", "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/"}]}