// Numbas version: finer_feedback_settings
{"name": "Probability, expectation and variance of exponential 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}, "variables": {"ans1": {"description": "", "templateType": "anything", "group": "Ungrouped variables", "definition": "precround(1/ra,3)", "name": "ans1"}, "period": {"description": "", "templateType": "anything", "group": "Ungrouped variables", "definition": "\"minutes\"", "name": "period"}, "ra": {"description": "", "templateType": "anything", "group": "Ungrouped variables", "definition": "random(0.2..1.2#0.1)", "name": "ra"}, "that": {"description": "", "templateType": "anything", "group": "Ungrouped variables", "definition": "\"two customers arriving \"", "name": "that"}, "thistime": {"description": "", "templateType": "anything", "group": "Ungrouped variables", "definition": "random(0.8..1.8#0.1)", "name": "thistime"}, "ans2": {"description": "", "templateType": "anything", "group": "Ungrouped variables", "definition": "precround(1/ra^2,3)", "name": "ans2"}, "tans3": {"description": "", "templateType": "anything", "group": "Ungrouped variables", "definition": "1-exp(-ra*thistime)", "name": "tans3"}, "ans3": {"description": "", "templateType": "anything", "group": "Ungrouped variables", "definition": "precround(tans3,3)", "name": "ans3"}, "this": {"description": "", "templateType": "anything", "group": "Ungrouped variables", "definition": "\"customer arrivals at the RyanJet check-in desk at Newcastle Airport \"", "name": "this"}, "tol": {"description": "", "templateType": "anything", "group": "Ungrouped variables", "definition": "0.001", "name": "tol"}}, "tags": ["checked2015", "MAS1403"], "name": "Probability, expectation and variance of exponential distribution", "advice": "\n    <p>If $X \\sim \\operatorname{exp}(\\lambda)$ then&nbsp;$\\displaystyle \\operatorname{E}[X] =\\frac{1}{\\lambda}$&nbsp;and &nbsp;$\\displaystyle \\operatorname{Var}(X)=\\frac{1}{\\lambda^2}$.</p>\n  <p>Also $P(X \\lt a)=1-e^{-\\lambda a}$.</p>\n  <p>a)</p>\n  <p>If $X \\sim \\operatorname{exp}(\\var{ra})$ then:</p>\n  <p>$\\displaystyle \\operatorname{E}[X] =\\frac{1}{\\lambda}=\\frac{1}{\\var{ra}}=\\var{ans1}$ to 3 decimal places.</p>\n  <p>$\\displaystyle \\operatorname{Var}(X) =\\frac{1}{\\lambda^2}=\\frac{1}{\\var{ra}^2}=\\var{ans2}$ to 3 decimal places.</p>\n  <p>b)</p>\n  <p>$P(X \\lt \\var{thistime}) = 1 -(e ^ {-\\var{ ra} \\times \\var{thistime}}) = 1 -(e ^ { -\\var{ra * thistime}}) = \\var{ans3}$ to 3 decimal places.</p>\n    ", "metadata": {"description": "<p>Question on the exponential distribution involving a time intervals and arrivals application, finding expectation and variance. Also finding the probability that a time interval between arrivals is less than a given period. All parameters and times randomised.&nbsp;</p>", "licence": "Creative Commons Attribution 4.0 International", "notes": "\n        \t\t<p><strong>1/01/2013:</strong></p>\n        \t\t    <p>This question can be changed to other applications via string variables. Added tag sc.</p>\n        \t\t"}, "statement": "\n    <p>The time, &nbsp;in {period} between {this} follows an exponential distribution with rate $\\var{ra}$ i.e.&nbsp;</p>\n  <p>\\[X \\sim \\operatorname{exp}(\\var{ra})\\]</p>\n    ", "variable_groups": [], "showQuestionGroupNames": false, "parts": [{"gaps": [{"allowFractions": false, "showCorrectAnswer": true, "type": "numberentry", "correctAnswerFraction": false, "showPrecisionHint": false, "minValue": "ans1-tol", "maxValue": "ans1+tol", "marks": 1, "scripts": {}}, {"allowFractions": false, "showCorrectAnswer": true, "type": "numberentry", "correctAnswerFraction": false, "showPrecisionHint": false, "minValue": "ans2-tol", "maxValue": "ans2+tol", "marks": 1, "scripts": {}}], "showCorrectAnswer": true, "type": "gapfill", "marks": 0, "scripts": {}, "prompt": "\n        <p>Find $\\operatorname{E}[X]$ between {this}:</p>\n      <p>$\\operatorname{E}[X]=\\;$?[[0]]{period} (enter as a decimal correct to 3 decimal places).</p>\n      <p>Find $\\operatorname{Var}(X)$:</p>\n      <p>$\\operatorname{Var}(X)=\\;$?[[1]](enter as a decimal correct to 3 decimal places).</p>\n        "}, {"gaps": [{"allowFractions": false, "showCorrectAnswer": true, "type": "numberentry", "correctAnswerFraction": false, "showPrecisionHint": false, "minValue": "ans3-tol", "maxValue": "ans3+tol", "marks": 1, "scripts": {}}], "showCorrectAnswer": true, "type": "gapfill", "marks": 0, "scripts": {}, "prompt": "<p>Find the probability that the time between {that} is less than $\\var{thistime}$ {period}:</p>\n        <p>&nbsp;$P(X \\lt \\var{thistime})=\\;$?[[0]](enter as a decimal correct to 3 decimal places)</p>"}], "ungrouped_variables": ["that", "this", "ans1", "ans2", "ans3", "period", "ra", "tol", "tans3", "thistime"], "type": "question", "preamble": {"js": "", "css": ""}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [], "pickQuestions": 0, "name": ""}], "functions": {}, "rulesets": {}, "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/"}], "resources": []}]}], "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/"}]}