// Numbas version: finer_feedback_settings
{"name": "Find cumulants with CGF", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"name": "Find cumulants with CGF", "tags": [], "metadata": {"description": "", "licence": "None specified"}, "statement": "<p>For CGF $\\log(m(s))$ what is the formula to find the $k^{\\mbox{th}}$ cumulant?</p>", "advice": "<p>This can easily be taken from the coefficients of $\\frac{s^k}{k!}$ without needing to derive.</p>\n<p>These are useful for finding the below:<br/>$\\begin{align*}<br/>k_1 &amp;= \\mu = E[X] \\\\<br/>k_2 &amp;= \\mu_2 = Var[X] \\\\<br/>k_3 &amp;= \\mu_3 = (3rd\\text{ central moment}) \\\\<br/>k_4 &amp;= \\mu_4 - 3 \\sigma^4 = (4th\\text{ central moment} - 3 \\sigma^4)<br/>\\end{align*}$</p>\n<p></p>", "rulesets": {}, "extensions": [], "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": "after"}}, "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": ["$\\kappa_k = \\frac{d^k\\log(m(s))}{ds^k}_{|s=0}$", "$\\kappa_k = \\frac{d^k\\log(m(s))}{ds^k}_{|s=1}$", "$\\kappa_k = \\frac{1}{k!} \\frac{d^k\\log(m(s))}{ds^k}_{|s=0}$"], "matrix": ["1", 0, 0], "distractors": ["", "", ""]}], "partsMode": "all", "maxMarks": 0, "objectives": [], "penalties": [], "objectiveVisibility": "always", "penaltyVisibility": "always", "type": "question", "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/"}]}