// Numbas version: finer_feedback_settings
{"name": "Find parameters of normal distributions from bivariate distribution, ", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"variable_groups": [], "variables": {"tol": {"templateType": "anything", "group": "Ungrouped variables", "definition": "0.01", "description": "", "name": "tol"}, "a": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(-5..5 except [0,1,-1])", "description": "", "name": "a"}, "c": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(-5..5 except [0,1,-1])", "description": "", "name": "c"}, "tansr": {"templateType": "anything", "group": "Ungrouped variables", "definition": "muy+rh*sigy*(f-mux)/sigx", "description": "", "name": "tansr"}, "en1": {"templateType": "anything", "group": "Ungrouped variables", "definition": "rh*sigx*sigy", "description": "", "name": "en1"}, "muy": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(1,2,3,4,5,10,20,50,100)", "description": "", "name": "muy"}, "ansp": {"templateType": "anything", "group": "Ungrouped variables", "definition": "precround(tansp,2)", "description": "", "name": "ansp"}, "tansq": {"templateType": "anything", "group": "Ungrouped variables", "definition": "sigx^2*(1-rh^2)", "description": "", "name": "tansq"}, "sigx": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(1,2,3,4,5,10)", "description": "", "name": "sigx"}, "b": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(-5..5 except [0,1,-1])", "description": "", "name": "b"}, "en2": {"templateType": "anything", "group": "Ungrouped variables", "definition": "sigx^2", "description": "", "name": "en2"}, "mux": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(1,2,3,4,5,10,20,50,100)", "description": "", "name": "mux"}, "d": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(-5..5 except [0,1,-1])", "description": "", "name": "d"}, "rh": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(-0.9..0.9#0.1)", "description": "", "name": "rh"}, "f": {"templateType": "anything", "group": "Ungrouped variables", "definition": "muy+random(-5..5 except 0)", "description": "", "name": "f"}, "tanss": {"templateType": "anything", "group": "Ungrouped variables", "definition": "sigy^2*(1-rh^2)", "description": "", "name": "tanss"}, "en3": {"templateType": "anything", "group": "Ungrouped variables", "definition": "sigy^2", "description": "", "name": "en3"}, "ansr": {"templateType": "anything", "group": "Ungrouped variables", "definition": "precround(tansr,2)", "description": "", "name": "ansr"}, "ansq": {"templateType": "anything", "group": "Ungrouped variables", "definition": "precround(tansq,2)", "description": "", "name": "ansq"}, "sigy": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(1,2,3,4,5,10)", "description": "", "name": "sigy"}, "g": {"templateType": "anything", "group": "Ungrouped variables", "definition": "mux+random(-5..5 except 0)", "description": "", "name": "g"}, "anss": {"templateType": "anything", "group": "Ungrouped variables", "definition": "precround(tanss,2)", "description": "", "name": "anss"}, "tansp": {"templateType": "anything", "group": "Ungrouped variables", "definition": "mux+rh*sigx*(g-muy)/sigy", "description": "", "name": "tansp"}}, "ungrouped_variables": ["sigy", "sigx", "tansq", "tansp", "tanss", "tansr", "muy", "mux", "tol", "rh", "ansp", "ansq", "ansr", "anss", "a", "c", "b", "d", "g", "f", "en1", "en2", "en3"], "question_groups": [{"pickingStrategy": "all-ordered", "questions": [], "name": "", "pickQuestions": 0}], "name": "Find parameters of normal distributions from bivariate distribution, ", "functions": {}, "showQuestionGroupNames": false, "parts": [{"scripts": {}, "gaps": [{"showCorrectAnswer": true, "allowFractions": false, "scripts": {}, "type": "numberentry", "maxValue": "a*mux", "minValue": "a*mux", "correctAnswerFraction": false, "marks": 0.5, "showPrecisionHint": false}, {"showCorrectAnswer": true, "allowFractions": false, "scripts": {}, "type": "numberentry", "maxValue": "a^2*sigx^2", "minValue": "a^2*sigx^2", "correctAnswerFraction": false, "marks": 0.5, "showPrecisionHint": false}], "type": "gapfill", "prompt": "<p>$\\var{a}X \\sim \\operatorname{N}(a,b)$ where:</p>\n      <p>$a=\\;$?[[0]] &nbsp; &nbsp;$b=\\;$?[[1]]</p>", "showCorrectAnswer": true, "marks": 0}, {"scripts": {}, "gaps": [{"showCorrectAnswer": true, "allowFractions": false, "scripts": {}, "type": "numberentry", "maxValue": "b*muy", "minValue": "b*muy", "correctAnswerFraction": false, "marks": 0.5, "showPrecisionHint": false}, {"showCorrectAnswer": true, "allowFractions": false, "scripts": {}, "type": "numberentry", "maxValue": "b^2*sigy^2", "minValue": "b^2*sigy^2", "correctAnswerFraction": false, "marks": 0.5, "showPrecisionHint": false}], "type": "gapfill", "prompt": "<p>$\\var{b}Y \\sim \\operatorname{N}(c,d)$ where:</p>\n      <p>$c=\\;$?[[0]] &nbsp; &nbsp;$d=\\;$?[[1]]</p>", "showCorrectAnswer": true, "marks": 0}, {"scripts": {}, "gaps": [{"showCorrectAnswer": true, "allowFractions": false, "scripts": {}, "type": "numberentry", "maxValue": "c*mux+d*muy", "minValue": "c*mux+d*muy", "correctAnswerFraction": false, "marks": 1, "showPrecisionHint": false}, {"showCorrectAnswer": true, "allowFractions": false, "scripts": {}, "type": "numberentry", "maxValue": "c^2*sigx^2+d^2*sigy^2+2*c*d*rh*sigx*sigy", "minValue": "c^2*sigx^2+d^2*sigy^2+2*c*d*rh*sigx*sigy", "correctAnswerFraction": false, "marks": 1, "showPrecisionHint": false}], "type": "gapfill", "prompt": "<p>$\\simplify{{c}X+{d}Y} \\sim \\operatorname{N}(f,g)$ where:</p>\n      <p>$f=\\;$?[[0]] &nbsp; &nbsp;$g=\\;$?[[1]]</p>", "showCorrectAnswer": true, "marks": 0}, {"scripts": {}, "gaps": [{"showCorrectAnswer": true, "allowFractions": false, "scripts": {}, "type": "numberentry", "maxValue": "ansp+tol", "minValue": "ansp-tol", "correctAnswerFraction": false, "marks": 1, "showPrecisionHint": false}, {"showCorrectAnswer": true, "allowFractions": false, "scripts": {}, "type": "numberentry", "maxValue": "ansq+tol", "minValue": "ansq-tol", "correctAnswerFraction": false, "marks": 1, "showPrecisionHint": false}, {"showCorrectAnswer": true, "allowFractions": false, "scripts": {}, "type": "numberentry", "maxValue": "ansr+tol", "minValue": "ansr-tol", "correctAnswerFraction": false, "marks": 1, "showPrecisionHint": false}, {"showCorrectAnswer": true, "allowFractions": false, "scripts": {}, "type": "numberentry", "maxValue": "anss+tol", "minValue": "anss-tol", "correctAnswerFraction": false, "marks": 1, "showPrecisionHint": false}], "type": "gapfill", "prompt": "<p>Input all answers in this part of the question to 2 decimal places.</p>\n      <p>$X|(Y=\\var{g}) \\sim&nbsp;\\operatorname{N}(p,q)$ &nbsp;where:</p>\n      <p>$p=\\;$?[[0]] &nbsp; &nbsp; $q=\\;$[[1]]</p>\n      <p>$Y|(X=\\var{f}) \\sim&nbsp;\\operatorname{N}(r,s)$ &nbsp;where:</p>\n      <p>$r=\\;$?[[2]] &nbsp; &nbsp; $s=\\;$[[3]]</p>", "showCorrectAnswer": true, "marks": 0}], "statement": "<p>Consider the following bivariate normal distribution.</p>\n  <p>\\[\\begin{pmatrix}X\\\\Y\\end{pmatrix} \\sim\\operatorname{N_2}\\left[\\begin{pmatrix}\\var{mux}\\\\ \\var{muy}\\end{pmatrix},\\begin{pmatrix} \\var{en2}&amp;\\var{en1}\\\\ \\var{en1}&amp;\\var{en3}\\end{pmatrix}\\right]\\]</p>\n  <p>You have to find the distributions of $\\var{a}X,\\;\\var{b}Y,\\;\\simplify{{c}X+{d}Y},\\; Y|(X=\\var{f}),\\;X|(Y=\\var{g})$</p>", "tags": ["bivariate normal ", "checked2015", "distributions", "linear combinations of normal distributions", "MAS1604", "MAS2304", "normal", "normal distributions", "normal parameters", "statistics"], "rulesets": {}, "preamble": {"css": "", "js": ""}, "type": "question", "metadata": {"notes": "<p><strong>7/02/2013:</strong></p>\n    <p>First draft finished.</p>", "licence": "Creative Commons Attribution 4.0 International", "description": "<p>Given the parameters of a bivariate Normal distribution $(X,Y)$ find the parameters of the Normal Distributions: $aX,\\;bY,\\;cX+dY,\\; Y|(X=f),\\;X|(Y=g)$</p>"}, "variablesTest": {"condition": "", "maxRuns": 100}, "advice": "<p>Look at your notes to see how these parameters for the distributions were obtained.</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/"}]}