// Numbas version: finer_feedback_settings
{"name": "20122013 CBA0_2", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"variable_groups": [], "variables": {"rs": {"group": "Ungrouped variables", "templateType": "anything", "definition": "[\n      ['$X_1$',random(2..6),random(1..5)],\n      ['$X_2$',random(2..6),random(1..5)],\n      ['$X_3$',random(2..6),random(1..5)]\n    ]", "name": "rs", "description": ""}, "var": {"group": "Ungrouped variables", "templateType": "anything", "definition": "a^2*rs[0][2]+b^2*rs[1][2]+c^2*rs[2][2]", "name": "var", "description": ""}, "d": {"group": "Ungrouped variables", "templateType": "anything", "definition": "random(-9..9 except [0,1,-1])", "name": "d", "description": ""}, "c": {"group": "Ungrouped variables", "templateType": "anything", "definition": "random(-9..9 except [0,1,-1])", "name": "c", "description": ""}, "b": {"group": "Ungrouped variables", "templateType": "anything", "definition": "random(-9..9 except [0,1,-1])", "name": "b", "description": ""}, "a": {"group": "Ungrouped variables", "templateType": "anything", "definition": "random(-9..9 except [0,1,-1])", "name": "a", "description": ""}, "ex": {"group": "Ungrouped variables", "templateType": "anything", "definition": "a*rs[0][1]+b*rs[1][1]+c*rs[2][1]+d", "name": "ex", "description": ""}}, "ungrouped_variables": ["a", "c", "b", "d", "rs", "ex", "var"], "rulesets": {}, "name": "20122013 CBA0_2", "showQuestionGroupNames": false, "functions": {}, "parts": [{"scripts": {}, "gaps": [{"scripts": {}, "showPrecisionHint": false, "allowFractions": false, "showCorrectAnswer": true, "type": "numberentry", "maxValue": "ex", "minValue": "ex", "correctAnswerFraction": false, "marks": 1}, {"scripts": {}, "showPrecisionHint": false, "allowFractions": false, "showCorrectAnswer": true, "type": "numberentry", "maxValue": "var", "minValue": "var", "correctAnswerFraction": false, "marks": 1}], "type": "gapfill", "prompt": "<p>Find the expectation and variance of:</p>\n      <p>\\[Y=\\simplify[basic]{{a}*X_1+{b}*X_2+{c}*X_3+{d}}\\]</p>\n      <p>$\\operatorname{E}[Y]=\\;$[[0]]&nbsp;</p>\n      <p>$\\operatorname{Var}(Y)=\\;$[[1]]</p>", "showCorrectAnswer": true, "marks": 0}], "statement": "<p>&nbsp;Suppose that the three independent random variables $X_1,\\;X_2,\\;X_3$ &nbsp;have the following distributions:</p>\n  <p>{table(rs,[' ','Mean','Variance'])}</p>", "tags": ["MAS2302", "checked2015"], "question_groups": [{"pickingStrategy": "all-ordered", "questions": [], "name": "", "pickQuestions": 0}], "preamble": {"css": "", "js": ""}, "type": "question", "metadata": {"notes": "<p><strong>25/01/2013:</strong></p>\n            <p>First draft completed. Needs testing.</p>", "licence": "Creative Commons Attribution 4.0 International", "description": "<p>Given a linear combination $Y$ of three independent random variables with given means and variances, find the mean of variance of $Y$.</p>"}, "variablesTest": {"condition": "", "maxRuns": 100}, "advice": "<p>If $Y=aX_1+bX_2+cX_3+d$ is a <strong>linear</strong> combination of <strong>independent&nbsp;</strong>&nbsp;random variables where $X_i$ &nbsp;has mean $m_i$ and variance $v_i,\\;i=1,\\;2,\\;3$ then&nbsp;</p>\n  <p>$\\operatorname{E}[Y]=am_1+bm_2+cm_3+d$ and $\\operatorname{Var}(Y)=a^2v_1+b^2v_2+c^2v_3$</p>\n  <p>For this example we have:</p>\n  <p>$\\operatorname{E}[Y]=\\simplify[basic]{{a}*{rs[0][1]}+{b}*{rs[1][1]}+{c}*{rs[2][1]}+{d}}=\\var{ex}$ and</p>\n  <p>$\\operatorname{Var}(Y)=\\simplify[basic]{{a^2}*{rs[0][2]}+{b^2}*{rs[1][2]}+{c^2}*{rs[2][2]}}=\\var{var}$.</p>\n  <p>&nbsp;</p>\n  <p>&nbsp;</p>", "contributors": [{"name": "Newcastle University Mathematics and Statistics", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/697/"}]}]}], "contributors": [{"name": "Newcastle University Mathematics and Statistics", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/697/"}]}