// Numbas version: exam_results_page_options
{"name": "The Exponential distribution", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"name": "The Exponential distribution", "tags": [], "metadata": {"description": "<p><em>No description</em></p>", "licence": "All rights reserved"}, "statement": "<p>Let the random variable $X$ have an Exponential distribution with rate parameter $\\lambda$:</p>\n<p style=\"padding-left: 40px;\">$X \\sim Exp(\\lambda)$</p>", "advice": "<p>The random variable $X\\sim Exp(\\lambda)$ has</p>\n<ul>\n<li>probability density function $f_X(x) = \\lambda e^{-\\lambda x}$ for $x&gt;0$ and $f_X(x)=0$ for $x\\le 0$</li>\n<li>cumulative distribution function $F_X(x) = 1 - e^{-\\lambda x}$ for $x&gt;0$ and $F_X(x)=0$ for $x\\le 0$</li>\n<li>$E[X] = 1/\\lambda$</li>\n<li>$Var[X] = \\frac{1}{\\lambda^2}$</li>\n<li>$Median[X] = \\log(2) / \\lambda$</li>\n<li>$P(X&gt;1) = e^{-\\lambda}$</li>\n</ul>", "rulesets": {}, "extensions": [], "builtin_constants": {"e": true, "pi,\u03c0": true, "i": true}, "constants": [], "variables": {"lambda": {"name": "lambda", "group": "Ungrouped variables", "definition": "random(1 .. 20#1)", "description": "<p>Rate parameter of the Exponential distribution.</p>", "templateType": "randrange", "can_override": true}, "lambda2": {"name": "lambda2", "group": "Ungrouped variables", "definition": "random(1 .. 20#1)", "description": "<p>Rate parameter of the Exponential distribution.</p>", "templateType": "randrange", "can_override": false}, "lambda3": {"name": "lambda3", "group": "Ungrouped variables", "definition": "random(1 .. 20#1)", "description": "", "templateType": "randrange", "can_override": false}, "lambda4": {"name": "lambda4", "group": "Ungrouped variables", "definition": "random(1 .. 5#0.5)", "description": "", "templateType": "randrange", "can_override": false}}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": ["lambda", "lambda2", "lambda3", "lambda4"], "variable_groups": [], "functions": {}, "preamble": {"js": "", "css": ""}, "parts": [{"type": "1_n_2", "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>The probability density function of $X$ is given by $f_X(x) =&nbsp; \\dots$ (for $x &gt; 0$)</p>", "minMarks": 0, "maxMarks": 0, "shuffleChoices": true, "displayType": "radiogroup", "displayColumns": 0, "showCellAnswerState": true, "choices": ["$\\lambda e^{-\\lambda x}$", "$e^{-\\lambda x}$", "$1 - e^{-\\lambda x}$", "$e^{\\lambda x}$"], "matrix": ["2", 0, "1", 0], "distractors": ["", "", "", ""]}, {"type": "1_n_2", "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>The cumulative distribution function of $X$ is given by $F_X(x)=&nbsp; \\dots $ (for $x &gt; 0$</p>", "minMarks": 0, "maxMarks": 0, "shuffleChoices": true, "displayType": "radiogroup", "displayColumns": 0, "showCellAnswerState": true, "choices": ["$\\lambda e^{-\\lambda x}$", "$\\lambda x$", "$1 - e^{-\\lambda x}$", "$1+e^{\\lambda x}$"], "matrix": ["1", "0", "2", 0], "distractors": ["", "", "", ""]}, {"type": "numberentry", "useCustomName": true, "customName": "Expectation", "marks": "2", "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<p>Calculate $E[X]$ when $\\lambda=$ {lambda}.</p>", "minValue": "1 / lambda", "maxValue": "1 / lambda", "correctAnswerFraction": true, "allowFractions": true, "mustBeReduced": false, "mustBeReducedPC": "50", "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}, {"type": "numberentry", "useCustomName": true, "customName": "Variance", "marks": "2", "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<p>Calculate $Var[X]$ when $\\lambda=$ {lambda2}.</p>", "minValue": "1 / lambda2^2", "maxValue": "1 / lambda2^2", "correctAnswerFraction": true, "allowFractions": true, "mustBeReduced": false, "mustBeReducedPC": "50", "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}, {"type": "numberentry", "useCustomName": true, "customName": "Median", "marks": "4", "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<p>Calculate the Median of $X$ when $\\lambda=$ {lambda3}.</p>", "minValue": "ln(2) / lambda3", "maxValue": "ln(2) / lambda3", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": "50", "precisionType": "sigfig", "precision": "2", "precisionPartialCredit": "75", "precisionMessage": "You have not given the correct number of significant figures.", "strictPrecision": false, "showPrecisionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}, {"type": "numberentry", "useCustomName": true, "customName": "Exceedance probability", "marks": "3", "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<p>Calculate $P(X &gt;1)$ when $\\lambda=$ {lambda4}.</p>", "minValue": "exp(-lambda4)", "maxValue": "exp(-lambda4)", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": "50", "precisionType": "sigfig", "precision": "2", "precisionPartialCredit": "65", "precisionMessage": "You have not given the correct number of significant figures.", "strictPrecision": false, "showPrecisionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}], "partsMode": "all", "maxMarks": 0, "objectives": [], "penalties": [], "objectiveVisibility": "always", "penaltyVisibility": "always", "contributors": [{"name": "Stefan Siegert", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/4982/"}]}]}], "contributors": [{"name": "Stefan Siegert", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/4982/"}]}