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{"name": "Find conditial transformation", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"name": "Find conditial transformation", "tags": [], "metadata": {"description": "", "licence": "None specified"}, "statement": "<p>Let $X$ and $Y$ be two continuous random variables with joint pdf $f(x, y)$.</p>\n<p>With $U=u(X,Y)$ and $V=v(X,Y)$ that are both 1-1 functions</p>\n<p>How can we find the conditional distributions $g(u|v)$ using no other information?</p>", "advice": "<p>To find the conditional distribution through a transformation we must find the joint and marginal to use in Bayes rule.</p>\n<ol>\n<li>First find the joint pdf $g(u,v)$ using a bivariate transformation</li>\n<li>Find the marginal $g(v) = \\int_Ug(u,v) du$</li>\n<li>Use Bayes rule to find the conditional $g(u|v) = \\frac{g(u,v)}{g(v)}$</li>\n</ol>", "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": "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": ["<ol>\n<li>First find the joint pdf $g(u,v)$ using a bivariate transformation</li>\n<li>Find the marginal $g(v) = \\int_Ug(u,v) du$</li>\n<li>Use Bayes rule to find the conditional $g(u|v) = \\frac{g(u,v)}{g(v)}$</li>\n</ol>", "<ol>\n<li>First find the joint pdf $g(u,v)$ using a bivariate transformation</li>\n<li>Find the marginal $g(u) = \\int_Vg(u,v) dv$</li>\n<li>Use Bayes rule to find the conditional $g(u|v) = \\frac{g(u,v)}{g(u)}$</li>\n</ol>", "<ol>\n<li>Find the conditional $f(x|y)$</li>\n<li>Transform $f(x|y) \\to g(u|v)$</li>\n</ol>"], "matrix": ["1", 0, 0], "distractors": ["", "", ""]}], "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/"}]}