// Numbas version: finer_feedback_settings
{"name": "PB3 - Expected Value and Standard Deviation for a Combination of Linear Random Variables", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"name": "PB3 - Expected Value and Standard Deviation for a Combination of Linear Random Variables", "tags": [], "metadata": {"description": "", "licence": "Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International"}, "statement": "<p>Two independent random variables $U$ and $V$ are used to compute a new random variable $W$ where $W = \\var{scalar}U + V$.</p>", "advice": "<p>In general if we have a linear combination of two independent random variables $X$ and $Y$, where $a$ and $b$ are constants, the formulas for expected value and variance are as follows:</p>\n<p>\\begin{align}<br/>E[aX + bY] &amp;= aE[X] + bE[Y] \\\\<br/>Var(aX+bY) &amp;= a^2Var(X) + b^2Var(Y).<br/>\\end{align}</p>\n<p><strong>a)</strong> We want to find the expected value of $W$, i.e., $E[W]$.</p>\n<p>Recall that $W = \\var{scalar}U + V.$</p>\n<p>Hence,&nbsp;</p>\n<p>\\begin{align}<br/>E[W] &amp;= \\var{scalar}E[U] + E[V] \\\\<br/>&amp;= \\var{scalar}*\\var{um} + \\var{vm} \\\\<br/>&amp;= \\var{scalar*um} + \\var{vm} \\\\<br/>&amp;= \\var{wm}.<br/>\\end{align}</p>\n<p>Hence the expected value of $W$ is $\\var{wm}$.</p>\n<p><strong>b)</strong> We want to find the standard deviation of $W$ ie., $\\sigma_W$. Let us first calculate the variance, i.e., $Var(W)$, since $\\sigma_W = \\sqrt{Var(W)}$.</p>\n<p>So,&nbsp;</p>\n<p>\\begin{align}<br/>Var(W) &amp;= \\var{scalar}^2Var(U) + Var(V).<br/>\\end{align}</p>\n<p>Here we need that $Var(U) = \\sigma_U^2 = \\var{usd}^2 = \\var{uvar}$ and $Var(V) = \\sigma_V^2 = \\var{vsd}^2 = \\var{vvar}$.</p>\n<p>So,</p>\n<p>\\begin{align}<br/>&amp;= \\var{scalar^2}\\cdot\\var{uvar} + \\var{vvar} \\\\<br/>&amp;= \\var{scalar^2*uvar} + \\var{vvar} \\\\<br/>&amp;= \\var{wvar}.<br/>\\end{align}</p>\n<p>We now need to calculate the standard deviation of $W$.</p>\n<p>So,&nbsp;</p>\n<p>\\begin{align}<br/>\\sigma_W &amp;= \\sqrt{Var(W)} \\\\<br/>&amp;= \\sqrt{\\var{wvar}} \\\\<br/>&amp;= \\var{wsd}.<br/>\\end{align}</p>\n<p>Hence, the standard deviation of $W$ is $\\var{round_wsd}$ to two decimal places.</p>\n<p></p>\n<p></p>", "rulesets": {}, "extensions": [], "builtin_constants": {"e": true, "pi,\u03c0": true, "i": true, "j": false}, "constants": [], "variables": {"um": {"name": "um", "group": "Ungrouped variables", "definition": "random(2,3,4,5,6,7,8,9,10)", "description": "<p>mean of u</p>", "templateType": "anything", "can_override": false}, "vm": {"name": "vm", "group": "Ungrouped variables", "definition": "random(2,3,4,5,6,7,8,9,10)", "description": "<p>mean of v</p>", "templateType": "anything", "can_override": false}, "scalar": {"name": "scalar", "group": "Ungrouped variables", "definition": "random(2,3,4,5)", "description": "<p>scalar multiplier of u in linear combination of random variables</p>", "templateType": "anything", "can_override": false}, "wm": {"name": "wm", "group": "Ungrouped variables", "definition": "scalar*um + vm", "description": "<p>mean of w</p>", "templateType": "anything", "can_override": false}, "usd": {"name": "usd", "group": "Ungrouped variables", "definition": "random(1,2,3,4,5)", "description": "<p>standard deviation of u</p>", "templateType": "anything", "can_override": false}, "vsd": {"name": "vsd", "group": "Ungrouped variables", "definition": "random(1,2,3,4,5)", "description": "<p>standard deviation of v</p>", "templateType": "anything", "can_override": false}, "wvar": {"name": "wvar", "group": "Ungrouped variables", "definition": "(scalar^2)*uvar + vvar", "description": "<p>variance of w</p>", "templateType": "anything", "can_override": false}, "wsd": {"name": "wsd", "group": "Ungrouped variables", "definition": "sqrt(wvar)", "description": "<p>standard deviation of w</p>", "templateType": "anything", "can_override": false}, "uvar": {"name": "uvar", "group": "Ungrouped variables", "definition": "usd^2", "description": "<p>variance of u</p>", "templateType": "anything", "can_override": false}, "vvar": {"name": "vvar", "group": "Ungrouped variables", "definition": "vsd^2", "description": "<p>variance of v</p>", "templateType": "anything", "can_override": false}, "round_wsd": {"name": "round_wsd", "group": "Ungrouped variables", "definition": "precround(wsd,2)", "description": "<p>standard deviation of w rounded to 2 dp</p>", "templateType": "anything", "can_override": false}}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": ["um", "vm", "scalar", "wm", "usd", "vsd", "wvar", "wsd", "uvar", "vvar", "round_wsd"], "variable_groups": [], "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>If the means of $U$ and $V$ are {um} and {vm}, respectively, what is the mean of $W$?</p>", "minValue": "wm", "maxValue": "wm", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "displayAnswer": "", "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}, {"type": "numberentry", "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>If the standard deviations of $U$ and $W$ are {usd} and {vsd}, resepctively, what is the standard deviation of $W$?</p>\n<p><em>If necessary, round your answer to two decimal places.</em></p>", "minValue": "round_wsd", "maxValue": "round_wsd", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "displayAnswer": "", "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}], "partsMode": "all", "maxMarks": 0, "objectives": [], "penalties": [], "objectiveVisibility": "always", "penaltyVisibility": "always", "contributors": [{"name": "Lauren Desoysa", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/21504/"}, {"name": "Megan Oliver", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/23526/"}], "resources": []}]}], "contributors": [{"name": "Lauren Desoysa", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/21504/"}, {"name": "Megan Oliver", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/23526/"}]}