// Numbas version: exam_results_page_options
{"name": "BS3.6", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"functions": {}, "ungrouped_variables": ["that", "this", "ans1", "ans2", "ans3", "period", "ra", "tol", "tans3", "thistime"], "name": "BS3.6", "tags": ["Probability", "continuous distributions", "distributions", "expectation", "exponential distribution", "probability", "sc", "statistical distributions", "statistics", "variance"], "preamble": {"css": "", "js": ""}, "advice": "<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<h4>a)</h4>\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<h4>b)</h4>\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>", "rulesets": {}, "parts": [{"prompt": "<p>Find $\\operatorname{E}[X]$ between {this}:</p>\n<p>$\\operatorname{E}[X]=$&nbsp;[[0]]{period} (enter as a decimal correct to 3 decimal places).</p>\n<p>Find $\\operatorname{Var}(X)$:</p>\n<p>$\\operatorname{Var}(X)=$&nbsp;[[1]]&nbsp;(enter as a decimal correct to 3 decimal places).</p>", "marks": 0, "gaps": [{"allowFractions": false, "marks": 1, "maxValue": "ans1+tol", "minValue": "ans1-tol", "correctAnswerFraction": false, "showCorrectAnswer": true, "scripts": {}, "type": "numberentry", "showPrecisionHint": false}, {"allowFractions": false, "marks": 1, "maxValue": "ans2+tol", "minValue": "ans2-tol", "correctAnswerFraction": false, "showCorrectAnswer": true, "scripts": {}, "type": "numberentry", "showPrecisionHint": false}], "showCorrectAnswer": true, "scripts": {}, "type": "gapfill"}, {"prompt": "<p>Find the probability that the time between {that} is less than $\\var{thistime}$ {period}:</p>\n<p>&nbsp;$P(X \\lt \\var{thistime})=$&nbsp;[[0]](enter as a decimal correct to 3 decimal places)</p>", "marks": 0, "gaps": [{"allowFractions": false, "marks": 1, "maxValue": "ans3+tol", "minValue": "ans3-tol", "correctAnswerFraction": false, "showCorrectAnswer": true, "scripts": {}, "type": "numberentry", "showPrecisionHint": false}], "showCorrectAnswer": true, "scripts": {}, "type": "gapfill"}], "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  \n    ", "variable_groups": [], "variablesTest": {"maxRuns": 100, "condition": ""}, "variables": {"that": {"definition": "\"two customers arriving \"", "templateType": "anything", "group": "Ungrouped variables", "name": "that", "description": ""}, "this": {"definition": "\"customer arrivals at the RyanJet check-in desk at Newcastle Airport \"", "templateType": "anything", "group": "Ungrouped variables", "name": "this", "description": ""}, "ans1": {"definition": "precround(1/ra,3)", "templateType": "anything", "group": "Ungrouped variables", "name": "ans1", "description": ""}, "ans2": {"definition": "precround(1/ra^2,3)", "templateType": "anything", "group": "Ungrouped variables", "name": "ans2", "description": ""}, "ans3": {"definition": "precround(tans3,3)", "templateType": "anything", "group": "Ungrouped variables", "name": "ans3", "description": ""}, "period": {"definition": "\"minutes\"", "templateType": "anything", "group": "Ungrouped variables", "name": "period", "description": ""}, "ra": {"definition": "random(0.2..1.2#0.1)", "templateType": "anything", "group": "Ungrouped variables", "name": "ra", "description": ""}, "tol": {"definition": "0.001", "templateType": "anything", "group": "Ungrouped variables", "name": "tol", "description": ""}, "tans3": {"definition": "1-exp(-ra*thistime)", "templateType": "anything", "group": "Ungrouped variables", "name": "tans3", "description": ""}, "thistime": {"definition": "random(0.8..1.8#0.1)", "templateType": "anything", "group": "Ungrouped variables", "name": "thistime", "description": ""}}, "metadata": {"notes": "\n        \t\t    \t\t<p><strong>1/01/2013:</strong></p>\n        \t\t    \t\t    <p>This question can be changed to other applications via string variables. Added tag sc.</p>\n        \t\t    \n        \t\t", "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"}, "type": "question", "showQuestionGroupNames": false, "question_groups": [{"name": "", "pickingStrategy": "all-ordered", "pickQuestions": 0, "questions": []}], "contributors": [{"name": "Bill Foster", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/6/"}]}]}], "contributors": [{"name": "Bill Foster", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/6/"}]}