// Numbas version: finer_feedback_settings
{"name": "Week 3 quiz", "metadata": {"description": "", "licence": "None specified"}, "duration": 0, "percentPass": 0, "showQuestionGroupNames": false, "shuffleQuestionGroups": true, "showstudentname": true, "question_groups": [{"name": "Group", "pickingStrategy": "all-shuffled", "pickQuestions": 1, "questionNames": ["", "", ""], "variable_overrides": [[], [], []], "questions": [{"name": "Range of RV transformation log", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Chris Drovandi", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/9840/"}, {"name": "adam bretherton", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/18177/"}], "tags": [], "metadata": {"description": "", "licence": "None specified"}, "statement": "<p>If $X$ has a range {a} &lt; $X$ &lt; {b} and $Y=\\log(X)$.</p>\n<p>What is the range of $Y$? (rounded to 3 decimal places)</p>", "advice": "", "rulesets": {}, "builtin_constants": {"e": true, "pi,\u03c0": true, "i": true}, "constants": [], "variables": {"a": {"name": "a", "group": "Ungrouped variables", "definition": "random(1 .. 10#1)", "description": "<p>Upper bound of Y</p>", "templateType": "randrange", "can_override": true}, "step": {"name": "step", "group": "Ungrouped variables", "definition": "random(1 .. 5#1)", "description": "<p>This make sure b &gt; a</p>", "templateType": "randrange", "can_override": false}, "b": {"name": "b", "group": "Ungrouped variables", "definition": "a + step", "description": "<p>Upper bound of X</p>", "templateType": "anything", "can_override": false}, "newb": {"name": "newb", "group": "Ungrouped variables", "definition": "max(ln(a), ln(b))", "description": "", "templateType": "anything", "can_override": false}, "newa": {"name": "newa", "group": "Ungrouped variables", "definition": "min(ln(a), ln(b))", "description": "", "templateType": 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"singleLetterVariables": false, "allowUnknownFunctions": true, "implicitFunctionComposition": false, "caseSensitive": false, "valuegenerators": []}], "sortAnswers": false}], "partsMode": "all", "maxMarks": 0, "objectives": [], "penalties": [], "objectiveVisibility": "always", "penaltyVisibility": "always", "type": "question"}, {"name": "Range of RV transformation power", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Chris Drovandi", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/9840/"}, {"name": "adam bretherton", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/18177/"}], "tags": [], "metadata": {"description": "", "licence": "None specified"}, "statement": "<p>If $X$ has a range {a} &lt; $X$ &lt; {b} and $Y=X^{\\var{c}}$.</p>\n<p>What is the range of $Y$?</p>", "advice": "", "rulesets": {}, "builtin_constants": {"e": true, "pi,\u03c0": true, "i": true}, "constants": [], "variables": {"a": {"name": "a", "group": "Ungrouped variables", "definition": "random(-10 .. 10#1)", "description": "<p>Upper bound of Y</p>", "templateType": "randrange", "can_override": true}, "step": {"name": "step", "group": "Ungrouped variables", "definition": "random(1 .. 5#1)", "description": "<p>This make sure b &gt; a</p>", "templateType": "randrange", "can_override": false}, "c": {"name": "c", "group": "Ungrouped variables", "definition": "random(2 .. 4#1)", "description": "<p>Scalar value for transformation Y=X^c</p>", "templateType": "randrange", "can_override": false}, "b": {"name": "b", "group": "Ungrouped variables", "definition": "a + step", "description": "<p>Upper bound of X</p>", "templateType": "anything", "can_override": false}, "newb": {"name": "newb", "group": "Ungrouped variables", "definition": "max(a^c, b^c)", "description": "", "templateType": "anything", "can_override": false}, 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"showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "answer": "{newa}", "showPreview": true, "checkingType": "dp", "checkingAccuracy": 3, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": false, "singleLetterVariables": false, "allowUnknownFunctions": true, "implicitFunctionComposition": false, "caseSensitive": false, "valuegenerators": []}, {"type": "jme", "useCustomName": false, "customName": "", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "answer": "{newb}", "showPreview": true, "checkingType": "dp", "checkingAccuracy": 3, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": false, "singleLetterVariables": false, "allowUnknownFunctions": true, "implicitFunctionComposition": false, "caseSensitive": false, "valuegenerators": []}], "sortAnswers": false}], "partsMode": "all", "maxMarks": 0, "objectives": [], "penalties": [], "objectiveVisibility": "always", "penaltyVisibility": "always", "type": "question"}, {"name": "Range of RV transformation scalar", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Chris Drovandi", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/9840/"}, {"name": "adam bretherton", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/18177/"}], "tags": [], "metadata": {"description": "", "licence": "None specified"}, "statement": "<p>If $X$ has a range {a} &lt; $X$ &lt; {b} and $Y=\\var{c}X$.</p>\n<p>What is the range of $Y$?</p>", "advice": "", "rulesets": {}, "builtin_constants": {"e": true, "pi,\u03c0": true, "i": true}, "constants": [], "variables": {"a": {"name": "a", "group": "Ungrouped variables", "definition": "random(-10 .. 10#1)", "description": "<p>Upper bound of Y</p>", "templateType": "randrange", "can_override": true}, "step": {"name": "step", "group": "Ungrouped variables", "definition": "random(1 .. 5#1)", "description": "<p>This make sure b &gt; a</p>", "templateType": "randrange", "can_override": false}, "c": {"name": "c", "group": "Ungrouped variables", "definition": "random(2 .. 5#1)", "description": "<p>Scalar value for transformation Y=cX</p>", "templateType": "randrange", "can_override": false}, "b": {"name": "b", "group": "Ungrouped variables", "definition": "a + step", "description": "<p>Upper bound of X</p>", "templateType": "anything", "can_override": false}, "newb": {"name": "newb", "group": "Ungrouped variables", "definition": "max(a*c, b*c)", "description": "", "templateType": "anything", "can_override": false}, "newa": {"name": "newa", "group": "Ungrouped variables", "definition": "min(a*c, a*c)", "description": "", "templateType": "anything", "can_override": false}}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": ["a", "step", "b", "c", "newa", "newb"], "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>[[0]] &lt; y &lt; [[1]]</p>", "gaps": [{"type": "jme", "useCustomName": false, "customName": "", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "answer": "{newa}", "showPreview": true, "checkingType": "dp", "checkingAccuracy": 3, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": false, "singleLetterVariables": false, "allowUnknownFunctions": true, "implicitFunctionComposition": false, "caseSensitive": false, "valuegenerators": []}, {"type": "jme", "useCustomName": false, "customName": "", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "answer": "{newb}", "showPreview": true, "checkingType": "dp", "checkingAccuracy": 3, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": false, "singleLetterVariables": false, "allowUnknownFunctions": true, "implicitFunctionComposition": false, "caseSensitive": false, "valuegenerators": []}], "sortAnswers": false}], "partsMode": "all", "maxMarks": 0, "objectives": [], "penalties": [], "objectiveVisibility": "always", "penaltyVisibility": "always", "type": "question"}]}, {"name": "Group", "pickingStrategy": "all-shuffled", "pickQuestions": 1, "questionNames": ["", "", "", "", "", "", "", "", ""], "variable_overrides": [[], [], [], [], [], [], [], [], []], "questions": [{"name": "Any kind of transformation", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "adam bretherton", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/18177/"}], "tags": [], "metadata": {"description": "", "licence": "None specified"}, "statement": "<p>Which of the following can be used to handle <strong>any</strong> kind of transformation?</p>", "advice": "<p>In principle any transform can be accommodated by the generic approach, although it might be difficult or impossible to do analytically.</p>", "rulesets": {}, "builtin_constants": {"e": true, "pi,\u03c0": true, "i": true}, "constants": [], "variables": {}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": [], "variable_groups": [], "functions": {}, "preamble": {"js": "", "css": ""}, "parts": [{"type": "1_n_2", "useCustomName": false, "customName": "", "marks": 0, "scripts": {"mark": {"script": "// Reveal answers, score and advice\nthis.display.revealAnswer();\nthis.display.showScore();\nNumbas.controls.getAdvice();", "order": "before"}}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minMarks": 0, "maxMarks": 0, "shuffleChoices": true, "displayType": "radiogroup", "displayColumns": 0, "showCellAnswerState": true, "choices": ["The generic transformation approach", "The 1-1 transformation approach", "Evaluating $f(\\alpha(x))$ where $\\alpha(\\cdot)$ is the transformation function and $f(x)$ is the pdf for $X$"], "matrix": ["1", 0, 0], "distractors": ["", "", ""]}], "partsMode": "all", "maxMarks": 0, "objectives": [], "penalties": [], "objectiveVisibility": "always", "penaltyVisibility": "always", "type": "question"}, {"name": "CDF for probability integral transform", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "adam bretherton", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/18177/"}], "tags": [], "metadata": {"description": "", "licence": "None specified"}, "statement": "<p>To apply the probability integral transform, the CDF of the random variable must be analytically available?</p>", "advice": "<p>To apply the probability integral transform, we only require that the <i data-stringify-type=\"italic\" style=\"box-sizing: inherit;\">quantile function</i>&nbsp;has an analytical form.&nbsp;</p>\n<p>There are some probability distributions, called quantile distributions, that are defined explicitly in terms of the their quantile function, and the corresponding CDF is not analytically tractable.</p>", "rulesets": {}, "builtin_constants": {"e": true, "pi,\u03c0": true, "i": true}, "constants": [], "variables": {}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": [], "variable_groups": [], "functions": {}, "preamble": {"js": "", "css": ""}, "parts": [{"type": "1_n_2", "useCustomName": false, "customName": "", "marks": 0, "scripts": {"mark": {"script": "// Reveal answers, score and advice\nthis.display.revealAnswer();\nthis.display.showScore();\nNumbas.controls.getAdvice();", "order": "before"}}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minMarks": 0, "maxMarks": 0, "shuffleChoices": true, "displayType": "radiogroup", "displayColumns": 0, "showCellAnswerState": true, "choices": ["False", "True"], "matrix": ["1", 0], "distractors": ["", ""]}], "partsMode": "all", "maxMarks": 0, "objectives": [], "penalties": [], "objectiveVisibility": "always", "penaltyVisibility": "always", "type": "question"}, {"name": "Chi-square and Normal transformation", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Chris Drovandi", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/9840/"}, {"name": "adam bretherton", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/18177/"}], "tags": [], "metadata": {"description": "", "licence": "None specified"}, "statement": "<p>Let $X \\sim N(0, 1)$ (standard Normal distribution)</p>\n<p>And $Y = X^2$</p>\n<p>Then $Y \\sim \\chi^2(1)$ (Chi-squared distribution with 1 degree of freedom)</p>\n<p>If $Z = \\sqrt{Y}$ why is $Z$ not the standard normal distribution?</p>", "advice": "<p>Taking the square of a standard normal produces a chi-square random variable.</p>\n<p>But taking the square root of the chi-square won't produce a normal since the range of the resulting distribution will be $(0, \\infty)$ not the entire set of real numbers and the PDF of $Z$ is $2$ times the PDF of $X$ so it still integrates to $1$.</p>\n<p>Performing both of these transformations is a good exercise.</p>", "rulesets": {}, "builtin_constants": {"e": true, "pi,\u03c0": true, "i": true}, "constants": [], "variables": {}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": [], "variable_groups": [], "functions": {}, "preamble": {"js": "", "css": ""}, "parts": [{"type": "m_n_2", "useCustomName": false, "customName": "", "marks": 0, "scripts": {"mark": {"script": "// Reveal answers, score and advice\nthis.display.revealAnswer();\nthis.display.showScore();\nNumbas.controls.getAdvice();", "order": "before"}}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minMarks": 0, "maxMarks": 0, "shuffleChoices": true, "displayType": "checkbox", "displayColumns": 0, "minAnswers": 0, "maxAnswers": 0, "warningType": "none", "showCellAnswerState": true, "markingMethod": "sum ticked cells", "choices": ["The range of $Z$ is $(0, \\infty)$ which is not unbounded like the Normal distribution requires.", "The PDF of $Z$ is different from the PDF of $X$", "The range of $Z$ is $(-\\pi, \\pi)$ which is not unbounded like the Normal distribution requires."], "matrix": ["1", "1", "-2"], "distractors": ["", "", ""]}], "partsMode": "all", "maxMarks": 0, "objectives": [], "penalties": [], "objectiveVisibility": "always", "penaltyVisibility": "always", "type": "question"}, {"name": "Find quantile functions", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "adam bretherton", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/18177/"}], "tags": [], "metadata": {"description": "", "licence": "None specified"}, "statement": "<p>Which of these distributions requires integration to compute their quantile function? </p>", "advice": "<p>The quantile function of the Normal distribution depends on the error function, which is an integral.</p>\n<p>The Exponential and Cauchy distributions have nice closed-form expressions for their quantile function.</p>", "rulesets": {}, "builtin_constants": {"e": true, "pi,\u03c0": true, "i": true}, "constants": [], "variables": {}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": [], "variable_groups": [], "functions": {}, "preamble": {"js": "", "css": ""}, "parts": [{"type": "m_n_2", "useCustomName": false, "customName": "", "marks": 0, "scripts": {"mark": {"script": "// Reveal answers, score and advice\nthis.display.revealAnswer();\nthis.display.showScore();\nNumbas.controls.getAdvice();", "order": "before"}}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minMarks": 0, "maxMarks": 0, "shuffleChoices": true, "displayType": "checkbox", "displayColumns": 0, "minAnswers": 0, "maxAnswers": 0, "warningType": "none", "showCellAnswerState": true, "markingMethod": "sum ticked cells", "choices": ["Normal", "Exponential", "Cauchy"], "matrix": ["1", "-1", "-1"], "distractors": ["", "", ""]}], "partsMode": "all", "maxMarks": 0, "objectives": [], "penalties": [], "objectiveVisibility": "always", "penaltyVisibility": "always", "type": "question"}, {"name": "Generating function transform", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "adam bretherton", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/18177/"}], "tags": [], "metadata": {"description": "", "licence": "None specified"}, "statement": "<p>We cannot use generating functions for handling transformations of random variables?</p>", "advice": "<p>Generating functions can be used for some simple transformations, such as sums of independent random variables.</p>", "rulesets": {}, "builtin_constants": {"e": true, "pi,\u03c0": true, "i": true}, "constants": [], "variables": {}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": [], "variable_groups": [], "functions": {}, "preamble": {"js": "", "css": ""}, "parts": [{"type": "1_n_2", "useCustomName": false, "customName": "", "marks": 0, "scripts": {"mark": {"script": "// Reveal answers, score and advice\nthis.display.revealAnswer();\nthis.display.showScore();\nNumbas.controls.getAdvice();", "order": "before"}}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minMarks": 0, "maxMarks": 0, "shuffleChoices": true, "displayType": "radiogroup", "displayColumns": 0, "showCellAnswerState": true, "choices": ["False", "True"], "matrix": ["1", 0], "distractors": ["", ""]}], "partsMode": "all", "maxMarks": 0, "objectives": [], "penalties": [], "objectiveVisibility": "always", "penaltyVisibility": "always", "type": "question"}, {"name": "Inverse of 1-1 transformation", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "adam bretherton", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/18177/"}], "tags": [], "metadata": {"description": "", "licence": "None specified"}, "statement": "<p>For every 1-1 transformation, there is a corresponding inverse transform?</p>", "advice": "<p>In the definition of 1-1 transformations, the transform must be invertible.</p>", "rulesets": {}, "builtin_constants": {"e": true, "pi,\u03c0": true, "i": true}, "constants": [], "variables": {}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": [], "variable_groups": [], "functions": {}, "preamble": {"js": "", "css": ""}, "parts": [{"type": "1_n_2", "useCustomName": false, "customName": "", "marks": 0, "scripts": {"mark": {"script": "// Reveal answers, score and advice\nthis.display.revealAnswer();\nthis.display.showScore();\nNumbas.controls.getAdvice();", "order": "before"}}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minMarks": 0, "maxMarks": 0, "shuffleChoices": true, "displayType": "radiogroup", "displayColumns": 0, "showCellAnswerState": true, "choices": ["True", "False"], "matrix": ["1", 0], "distractors": ["", ""]}], "partsMode": "all", "maxMarks": 0, "objectives": [], "penalties": [], "objectiveVisibility": "always", "penaltyVisibility": "always", "type": "question"}, {"name": "Jacobian inverse", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Chris Drovandi", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/9840/"}, {"name": "adam bretherton", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/18177/"}], "tags": [], "metadata": {"description": "", "licence": "None specified"}, "statement": "<p>To compute the Jacobian term for 1-1 transformations, we must differentiate the inverse transform?</p>", "advice": "<p>We can in principle avoid differentiating the inverse transform by&nbsp; using the result that $\\frac{dx}{dy} = 1/\\frac{dy}{dx}$, where the RHS only requires the forward transformation.&nbsp; This is nice to do when differentiating the inverse transform is difficult.</p>", "rulesets": {}, "builtin_constants": {"e": true, "pi,\u03c0": true, "i": true}, "constants": [], "variables": {}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": [], "variable_groups": [], "functions": {}, "preamble": {"js": "", "css": ""}, "parts": [{"type": "1_n_2", "useCustomName": false, "customName": "", "marks": 0, "scripts": {"mark": {"script": "// Reveal answers, score and 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false}, "contributors": [{"name": "Chris Drovandi", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/9840/"}, {"name": "adam bretherton", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/18177/"}], "tags": [], "metadata": {"description": "", "licence": "None specified"}, "statement": "<p>We can use the probability integral transform to sample from distributions based on uniform random numbers?</p>", "advice": "<p>Being able to simulate random numbers from distributions based on uniform random numbers is one of the main benefits of the probability integral transform.</p>", "rulesets": {}, "builtin_constants": {"e": true, "pi,\u03c0": true, "i": true}, "constants": [], "variables": {}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": [], "variable_groups": [], "functions": {}, "preamble": {"js": "", "css": ""}, "parts": [{"type": "1_n_2", "useCustomName": false, "customName": "", "marks": 0, "scripts": {"mark": {"script": "// 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