// Numbas version: exam_results_page_options {"name": "Interactive Brownian motion", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"name": "Interactive Brownian motion", "tags": [], "metadata": {"description": "", "licence": "None specified"}, "statement": "

Consider $W(t)$ a standard Brownian motion at two time time points $\\var{s}$ and $\\var{t}$.

\n

Find the following:

", "advice": "

Remember that a standard Brownian motion has $\\sigma^2 = 1$ and that $W(t) - W(s) \\sim N(0, t-s)$ when $t > s$.

$E[W(\\var{s}) - W(\\var{t})] = E[W(\\var{s})] - E[W(\\var{t})] = 0 - 0 = 0$

\n

$Var[W(\\var{t}) - W(\\var{s})] = \\var{t}-\\var{s}$

\n

$Cov[W(\\var{s}),W(\\var{t})] : Cov[W(s), W(t)] = min(s, t) $

\n

$E[W(\\var{s})W(\\var{t})]: E[W(s)W(t)]  = Cov[W(s), W(t)] + E[W(s)]E[W(t)] = Cov[W(s)W(t)] + 0 = min(s, t)$

", "rulesets": {}, "extensions": [], "builtin_constants": {"e": true, "pi,\u03c0": true, "i": true}, "constants": [], "variables": {"s": {"name": "s", "group": "Ungrouped variables", "definition": "random(1..10)", "description": "", "templateType": "anything", "can_override": false}, "t": {"name": "t", "group": "Ungrouped variables", "definition": "s+random(1..10)", "description": "", "templateType": "anything", "can_override": false}}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": ["s", "t"], "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": "

$E[W(\\var{s}) - W(\\var{t})] = $ [[0]]

\n

$Var[W(\\var{t}) - W(\\var{s})] = $ [[1]]

\n

$Cov[W(\\var{s}),W(\\var{t})] = $ [[2]]

\n

$E[W(\\var{s})W(\\var{t})] = $ [[3]]

", "gaps": [{"type": "jme", "useCustomName": false, "customName": "", "marks": 1, "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, "answer": "0", "showPreview": true, "checkingType": "absdiff", "checkingAccuracy": 0.001, "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": {"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, "answer": "{t}-{s}", "showPreview": true, "checkingType": "absdiff", "checkingAccuracy": 0.001, "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": {"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, "answer": "{s}", "showPreview": true, "checkingType": "absdiff", "checkingAccuracy": 0.001, "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": {"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, "answer": "{s}", "showPreview": true, "checkingType": "absdiff", "checkingAccuracy": 0.001, "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", "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/"}]}]}], "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/"}]}