// Numbas version: exam_results_page_options
{"name": "Probability: Discrete Random Variable 2", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"name": "Probability: Discrete Random Variable 2", "tags": [], "metadata": {"description": "", "licence": "None specified"}, "statement": "<p>Given a discrete random variable (in a table) calculate:<br/>E(X), Var, std dev, cumulative distribution and probabilties.</p>", "advice": "<h3>part a)</h3>\n<p>A probability distribution is a function that defines the probability of each possible outcome. &nbsp;Since the sum of probabilities of all outcomes must be 1, we can use this fact to calculate the missing probability.<br/><br/>Therefore <br/>\\[&nbsp;<br/>P(\\var{X[0]} = \\var{x1}) +&nbsp;P(\\var{X[0]}&nbsp; = \\var{x2})&nbsp;+&nbsp;P(\\var{X[0]}=\\var{x3}) + P(\\var{X[0]}=\\var{x4}) +P(\\var{X[0]}=\\var{x5}) = 1&nbsp;\\\\<br/>\\implies &nbsp;\\var{p1} + &nbsp;P(\\var{X[0]} = \\var{x2}) + \\var{p3} + \\var{p4} + \\var{p5} = 1&nbsp;&nbsp;\\]<br/><br/>Which we can solve to show that&nbsp;</p>\n<p>\\[&nbsp;P(\\var{X[0]} = \\var{x2}) = \\var{p2}\\]</p>\n<h3><br/>part b)</h3>\n<p>The expected value is calculated by multiplying each value by it's probability, and adding these up. &nbsp;<br/><br/>You can use the formula:<br/><br/>\\[ E(\\var{X[0]}) = \\sum_{\\var{X[1]}=1}^{5}&nbsp;&nbsp;\\var{X[1]} \\times &nbsp;P(\\var{X[0]} = \\var{X[1]}) \\]<br/><br/>Which gives:<br/><br/>\\[ \\begin{split}<br/>E(\\var{X[0]}) &amp;= \\var{x1} \\times \\var{p1} + \\var{x2} \\times \\var{p2} + \\var{x3} \\times \\var{p3} + \\var{x4} \\times \\var{p4} + \\var{x5} \\times \\var{p5} \\\\ &amp;= \\var{ex}<br/>\\end{split} \\]<br/><br/>The expectation can be understood to be the average&nbsp;value you will get if you select a number from the distribution many, many times. &nbsp;Therefore it makes sense for the expectation to be roughly in the middle of the $\\var{x[1]}$ values.</p>\n<h3>part c) &nbsp;</h3>\n<p>The variance is calculated using the formula:<br/><br/>\\[ Var(\\var{X[0]}) = E \\left(\\var{X[0]}^2\\right) - [E(\\var{X[0]})]^2\\]<br/><br/>Which you can remember by saying 'the expectation of the square subtract the square of the expectation'.<br/><br/>We worked out $E(\\var{X[0]})$ in part b), but we need to find $E \\left(\\var{X[0]}^2 \\right)$, so we need to square the values before performing the expectation calculation:<br/><br/>\\[E \\left(\\var{X[0]}^2 \\right) = \\sum_{\\var{X[1]}=1}^{5} &nbsp;\\var{X[1]}^2 \\times &nbsp;P(\\var{X[0]} = \\var{X[1]}) \\]</p>\n<p><br/>Which gives:<br/><br/>\\[ \\begin{split}<br/>E \\left(\\var{X[0]}^2 \\right) &amp;= \\var{x1^2} \\times \\var{p1} + \\var{x2^2} \\times \\var{p2} + \\var{x3^2} \\times \\var{p3} + \\var{x4^2} \\times \\var{p4} + \\var{x5^2} \\times \\var{p5} \\\\ &amp;= \\var{exsquared} <br/>\\end{split} \\]<br/><br/>Therefore<br/><br/></p>\n<p>{partc}</p>\n<h3>part d)</h3>\n<p>The standard deviation is the square root of the variance, so not too much work to do for this one, phewph! &nbsp;</p>\n<p>To avoid losing accuracy in your answer for the standard deviation, you should use the unrounded answer for the variance you calculated in part c).</p>\n<p>{partd}</p>\n<h3>part e)</h3>\n<p>The cumulative distribution function can be thought of as a 'less than this value' probability distribution. &nbsp;</p>\n<p>\\[F(\\var{x[1]}) =&nbsp;P(\\var{X[0]} \\le&nbsp;\\var{x[1]})\\].<br/><br/>So if we take $P(\\var{X[0]} \\le&nbsp;\\var{x2})$ as an example, we can see that&nbsp;<br/><br/>\\[P(\\var{X[0]} \\le&nbsp;\\var{x2}) =&nbsp;P(\\var{X[0]} =&nbsp;\\var{x1}) +&nbsp;P(\\var{X[0]} =&nbsp;\\var{x2})\\]</p>\n<h3>part f)</h3>\n<p>To find $P(\\var{xset[l-1]} &lt; \\var{X[0]} \\le \\var{xset[u-1]})$ we can use the cumulative frequency table to help us. &nbsp;Remember that this is a discrete random 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"group": "Ungrouped variables", "definition": "\"<p>\\\\[ \\\\begin{split}<br /><br />\\\\text{Standard deviation} &amp;\\\\,= \\\\sqrt{Var(\\\\var{X[0]})} \\\\\\\\ &amp;\\\\,= \\\\sqrt{\\\\var{varx}} \\\\\\\\ &amp;\\\\,= \\\\var{std}<br /><br />\\\\end{split} \\\\]</p>\"", "description": "", "templateType": "long string", "can_override": false}, "sold2": {"name": "sold2", "group": "Ungrouped variables", "definition": "\"<p>\\\\[ \\\\begin{split}<br /><br />\\\\text{Standard deviation} &amp;\\\\,= \\\\sqrt{Var(\\\\var{X[0]})} \\\\\\\\ &amp;\\\\,= \\\\sqrt{\\\\var{varx}} \\\\\\\\ &amp;\\\\,= \\\\var{std} \\\\\\\\ &amp;\\\\,= \\\\var{std2} \\\\text{ (2 d.p.)}<br /><br />\\\\end{split} \\\\]</p>\"", "description": "", "templateType": "long string", "can_override": false}, "std": {"name": "std", "group": "Ungrouped variables", "definition": "sqrt(varx)", "description": "", "templateType": "anything", "can_override": false}, "partc": {"name": "partc", "group": "Ungrouped variables", "definition": "if(abs(varx-varx2)<0.001,'{solc1}','{solc2}')", "description": "", "templateType": "anything", "can_override": false}, "solc2": {"name": "solc2", "group": "Ungrouped variables", "definition": "\"<p>\\\\[ \\\\begin{split} <br />Var(\\\\var{X[0]}) &amp;= \\\\var{exsquared} - \\\\var{ex^2} \\\\\\\\ &amp;= \\\\var{varx} \\\\\\\\ &amp;=\\\\var{varx2} \\\\text{ (2 d.p.)}<br />\\\\end{split} \\\\]</p>\"", "description": "", "templateType": "long string", "can_override": false}, "solc1": {"name": "solc1", "group": "Ungrouped variables", "definition": "\"<p>\\\\[ \\\\begin{split}<br />Var(\\\\var{X[0]}) &amp;= \\\\var{exsquared} - \\\\var{ex^2} \\\\\\\\ &amp;= \\\\var{varx}\\\\end{split} \\\\]</p>\"", "description": "", "templateType": "long string", "can_override": false}, "varx2": {"name": "varx2", "group": "Unnamed group", "definition": "precround(varx,2)", "description": "", "templateType": "anything", "can_override": false}}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": ["X", "n", "p1", "p2", "p3", "p4", "p5", "CHECKSUM", "ex", "exsquared", "x3", "x4", "x5", "x1", "xset", "xsets", "x2", "xsetchoice", "std", "std2", "cum", "l", "u", "lastpart", "partd", "sold1", "sold2", "partc", "solc1", "solc2"], "variable_groups": [{"name": "Unnamed group", "variables": ["varx", "varx2"]}], "functions": {}, "preamble": {"js": "", "css": ""}, "parts": [{"type": "gapfill", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<p></p>\n<p>The probability distribution of a discrete random variable $\\var{X[0]}$ is given by:</p>\n<table border=\"yes\" cellpadding=\"10\" cellspacing=\"10\">\n<tbody>\n<tr>\n<td>$\\var{x[1]}$</td>\n<td><br/>&nbsp; &nbsp;$\\var{X1}$ &nbsp;&nbsp;<br/><br/></td>\n<td>&nbsp; &nbsp;$\\var{X2}$ &nbsp;</td>\n<td>&nbsp; &nbsp;$\\var{X3}$ &nbsp;</td>\n<td>&nbsp; &nbsp;$\\var{X4}$ &nbsp;</td>\n<td>&nbsp; &nbsp;$\\var{X5}$ &nbsp;</td>\n</tr>\n<tr>\n<td>$f(\\var{x[1]})$<br/>= $P(\\var{X[0]} =&nbsp;\\var{x[1]})$</td>\n<td><br/>&nbsp; &nbsp;$\\var{p1}$ <br/>&nbsp;</td>\n<td>&nbsp; &nbsp;[[0]]</td>\n<td>&nbsp; &nbsp;$\\var{p3}$ &nbsp;</td>\n<td>&nbsp; &nbsp;$\\var{p4}$ &nbsp;</td>\n<td>$\\var{p5}$ &nbsp;</td>\n</tr>\n</tbody>\n</table>", "gaps": [{"type": "numberentry", "useCustomName": false, "customName": "", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "p2", "maxValue": "p2", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "displayAnswer": "", "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}], "sortAnswers": false}, {"type": "numberentry", "useCustomName": false, "customName": "", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<p>Calculate the expected value, E($\\var{X[0]}$)</p>", "minValue": "ex", "maxValue": "ex", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "displayAnswer": "", "precisionType": "dp", "precision": "2", 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"variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "answer": "{precround(varx,2)}", "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": []}], "sortAnswers": false}, {"type": "numberentry", "useCustomName": false, "customName": "", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<p>Calculate the standard deviation of $\\var{x[0]}$.</p>", "minValue": "floor(std2)", "maxValue": "ceil(std2)", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "displayAnswer": "", "precisionType": "dp", "precision": "2", "precisionPartialCredit": 0, "precisionMessage": "You have not given your answer to the correct precision.", "strictPrecision": false, "showPrecisionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}, {"type": "gapfill", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<p>Find the cumulative distribution function, $F(\\var{x[1]})$,<br/><br/></p>\n<table style=\"border-color: #000000; float: left;\" border=\"yes\" cellpadding=\"10\" cellspacing=\"10\">\n<tbody>\n<tr>\n<td style=\"text-align: center; width: 214.421875px;\">$\\var{x[1]}$</td>\n<td style=\"text-align: center; width: 112.359375px;\"><br/>&nbsp; &nbsp;$\\var{X1}$ &nbsp;&nbsp;<br/><br/></td>\n<td style=\"text-align: center; width: 107.4375px;\">&nbsp; &nbsp;$\\var{X2}$ &nbsp;</td>\n<td style=\"text-align: center; width: 107.4375px;\">&nbsp; &nbsp;$\\var{X3}$ &nbsp;</td>\n<td style=\"text-align: center; width: 107.4375px;\">&nbsp; &nbsp;$\\var{X4}$ &nbsp;</td>\n<td style=\"text-align: center; width: 107.4375px;\">&nbsp; &nbsp;$\\var{X5}$ &nbsp;</td>\n</tr>\n<tr>\n<td style=\"text-align: center; width: 214.421875px;\">$F(\\var{x[1]})$<br/>= $P(\\var{X[0]} \\le&nbsp;\\var{x[1]})$</td>\n<td style=\"text-align: center; width: 112.359375px;\"><br/>&nbsp; &nbsp;&nbsp;[[0]]<br/>&nbsp;</td>\n<td style=\"text-align: center; width: 107.4375px;\">[[1]]&nbsp; &nbsp; &nbsp;</td>\n<td style=\"text-align: center; width: 107.4375px;\">[[2]]&nbsp; &nbsp;&nbsp;</td>\n<td style=\"text-align: center; width: 107.4375px;\">&nbsp; [[3]]&nbsp;&nbsp;</td>\n<td style=\"text-align: center; width: 107.4375px;\">[[4]]</td>\n</tr>\n</tbody>\n</table>", "gaps": [{"type": "numberentry", "useCustomName": false, "customName": "", "marks": "0.2", "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "cum[0]", "maxValue": "cum[0]", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "displayAnswer": "", "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}, {"type": "numberentry", "useCustomName": false, "customName": "", "marks": "0.2", 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