// Numbas version: exam_results_page_options
{"name": "PDF and CDF of continuous uniform random variable, ", "extensions": ["stats"], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"variable_groups": [], "variables": {"a": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(-5..5)", "name": "a", "description": ""}, "x1": {"templateType": "anything", "group": "Ungrouped variables", "definition": "a+random(1..4)", "name": "x1", "description": ""}, "b": {"templateType": "anything", "group": "Ungrouped variables", "definition": "a+random(5..12)", "name": "b", "description": ""}}, "ungrouped_variables": ["a", "x1", "b"], "rulesets": {"std": ["all", "fractionNumbers", "!collectNumbers", "!noLeadingMinus"]}, "name": "PDF and CDF of continuous uniform random variable, ", "functions": {}, "showQuestionGroupNames": false, "parts": [{"scripts": {}, "gaps": [{"answer": "1/{b-a}", "showCorrectAnswer": true, "vsetrange": [0, 1], "scripts": {}, "checkvariablenames": false, "expectedvariablenames": [], "notallowed": {"message": "<p>input as a fraction and not a decimal</p>", "showStrings": false, "partialCredit": 0, "strings": ["."]}, "showpreview": true, "checkingtype": "absdiff", "checkingaccuracy": 0.001, "type": "jme", "answersimplification": "std", "marks": 0.5, "vsetrangepoints": 5}, {"answer": "0", "vsetrange": [0, 1], "scripts": {}, "checkvariablenames": false, "expectedvariablenames": [], "notallowed": {"message": "<p>Input as a fraction or an integer.</p>", "showStrings": false, "partialCredit": 0, "strings": ["."]}, "showpreview": true, "checkingtype": "absdiff", "checkingaccuracy": 0.001, "type": "jme", "showCorrectAnswer": true, "marks": 0.5, "vsetrangepoints": 5}], "type": "gapfill", "showCorrectAnswer": true, "prompt": "<p>&nbsp;</p>\n        <p>What is the PDF of $X$? Input all answers as fractions or integers, not as decimals.</p>\n        <table>\n        <tbody>\n        <tr>\n        <td rowspan=\"7\">$f_X(x) = \\left \\{ \\begin{array}{l} \\phantom{{.}} \\\\\\phantom{{.}} \\\\ \\phantom{{.}} \\\\ \\phantom{{.}}\\end{array} \\right .$</td>\n        <td style=\"text-align: left;\">[[0]]</td>\n        <td>$\\var{a} \\leq x \\leq \\var{b},$</td>\n        </tr>\n        <tr>\n        <td style=\"text-align: left;\">[[1]]</td>\n        <td>otherwise</td>\n        </tr>\n        </tbody>\n        </table>\n        <p>&nbsp;</p>", "marks": 0}, {"scripts": {}, "gaps": [{"answer": "0", "vsetrange": [0, 1], "checkingaccuracy": 0.001, "checkvariablenames": false, "expectedvariablenames": [], "showpreview": true, "checkingtype": "absdiff", "scripts": {}, "type": "jme", "showCorrectAnswer": true, "marks": 0.5, "vsetrangepoints": 5}, {"answer": "(x-{a})/({b-a})", "showCorrectAnswer": true, "vsetrange": [0, 1], "scripts": {}, "checkvariablenames": false, "expectedvariablenames": [], "notallowed": {"message": "<p>input numbers as fractions or integers and not as decimals</p>", "showStrings": false, "partialCredit": 0, "strings": ["."]}, "showpreview": true, "checkingtype": "absdiff", "checkingaccuracy": 0.001, "type": "jme", "answersimplification": "std", "marks": 1, "vsetrangepoints": 5}, {"answer": "1", "showCorrectAnswer": true, "vsetrange": [0, 1], "scripts": {}, "checkvariablenames": false, "expectedvariablenames": [], "notallowed": {"message": "<p>input numbers as fractions or integers and not as decimals</p>", "showStrings": false, "partialCredit": 0, "strings": ["."]}, "showpreview": true, "checkingtype": "absdiff", "checkingaccuracy": 0.001, "type": "jme", "answersimplification": "std", "marks": 0.5, "vsetrangepoints": 5}], "type": "gapfill", "showCorrectAnswer": true, "prompt": "<p>Compute the CDF $F_X(x)$ of $X$.</p>\n        <table>\n        <tbody>\n        <tr>\n        <td rowspan=\"7\">$F_X(x) = \\left \\{ \\begin{array}{l} \\phantom{{.}} \\\\ \\phantom{{.}} \\\\ \\phantom{{.}} \\\\ \\phantom{{.}} \\\\ \\phantom{{.}}\\\\ \\phantom{{.}}\\\\ \\phantom{{.}} \\end{array} \\right .$</td>\n        <td style=\"text-align: left;\">[[0]]</td>\n        <td>$x \\lt \\var{a},$</td>\n        </tr>\n        <tr>\n        <td>&nbsp;</td>\n        <td>&nbsp;</td>\n        </tr>\n        <tr>\n        <td style=\"text-align: left;\">[[1]]</td>\n        <td>$\\var{a} \\leq x \\leq \\var{b}$</td>\n        </tr>\n        <tr>\n        <td>&nbsp;</td>\n        <td>&nbsp;</td>\n        </tr>\n        <tr>\n        <td style=\"text-align: left;\">[[2]]</td>\n        <td>$ x \\gt \\var{b},$</td>\n        </tr>\n        <tr>\n        <td>&nbsp;</td>\n        <td>&nbsp;</td>\n        </tr>\n        </tbody>\n        </table>\n        <p>Input all numbers as fractions or integers in the above formulae.</p>", "marks": 0}, {"scripts": {}, "gaps": [{"answer": "{b-x1}/{b-a}", "showCorrectAnswer": true, "vsetrange": [0, 1], "scripts": {}, "checkvariablenames": false, "expectedvariablenames": [], "notallowed": {"message": "<p>Input all numbers as fractions or integers.</p>", "showStrings": false, "partialCredit": 0, "strings": ["."]}, "showpreview": true, "checkingtype": "absdiff", "checkingaccuracy": 0.001, "type": "jme", "answersimplification": "std", "marks": 1, "vsetrangepoints": 5}], "type": "gapfill", "showCorrectAnswer": true, "prompt": "<p>Also, using the distribution function above find:</p>\n        <p>$P( X \\ge \\var{x1})=\\;\\;$[[0]]</p>\n        <p>(input your answer as a fraction or integer and not as a decimal).</p>", "marks": 0}], "statement": "<p>Suppose $X$ is a continuous uniform random variable defined on $[\\var{a},\\;\\var{b}]$.</p>", "tags": ["CFD", "checked2015", "continuous random variables", "cumulative distribution functions", "density functions", "distribution function", "distribution functions", "MAS1604", "MAS2304", "pdf", "PDF", "Probability", "probability density function", "random variables", "statistics", "uniform random variable"], "question_groups": [{"pickingStrategy": "all-ordered", "questions": [], "name": "", "pickQuestions": 0}], "preamble": {"css": "", "js": ""}, "type": "question", "metadata": {"notes": "<p><strong>05/02/2013:</strong></p>\n        <p>First draft finished.&nbsp;</p>", "licence": "Creative Commons Attribution 4.0 International", "description": "<p>$X$ is a continuous uniform random variable defined on $[a,\\;b]$. Find the PDF and CDF of $X$ and find &nbsp;$P(X \\ge c)$.</p>"}, "variablesTest": {"condition": "", "maxRuns": 100}, "advice": "<p>a)</p>\n    <p>The PDF is given by:</p>\n    <p>&nbsp;</p>\n    <table>\n    <tbody>\n    <tr>\n    <td rowspan=\"7\">$f_X(x) = \\left \\{ \\begin{array}{l} \\phantom{{.}} \\\\\\phantom{{.}} \\\\ \\phantom{{.}} \\\\ \\phantom{{.}}\\end{array} \\right .$</td>\n    <td style=\"text-align: left;\">\\[\\frac{1}{\\var{b}-(\\var{a})}=\\frac{1}{\\var{b-a}}\\]</td>\n    <td>$\\var{a} \\leq x \\leq \\var{b},$</td>\n    </tr>\n    <tr>\n    <td style=\"text-align: left;\">$0$</td>\n    <td>otherwise</td>\n    </tr>\n    </tbody>\n    </table>\n    <p>b) The CDF is given by:</p>\n    <table>\n    <tbody>\n    <tr>\n    <td rowspan=\"7\">$F_X(x) = \\left \\{ \\begin{array}{l} \\phantom{{.}} \\\\ \\phantom{{.}} \\\\ \\phantom{{.}} \\\\ \\phantom{{.}} \\\\ \\phantom{{.}}\\\\ \\phantom{{.}}\\\\ \\phantom{{.}} \\end{array} \\right .$</td>\n    <td style=\"text-align: left;\">$0$</td>\n    <td>$x \\lt \\var{a},$</td>\n    </tr>\n    <tr>\n    <td>&nbsp;</td>\n    <td>&nbsp;</td>\n    </tr>\n    <tr>\n    <td style=\"text-align: left;\">\\[\\simplify{(x-{a})/{b-a}}\\]</td>\n    <td>$\\var{a} \\leq x \\leq \\var{b}$</td>\n    </tr>\n    <tr>\n    <td>&nbsp;</td>\n    <td>&nbsp;</td>\n    </tr>\n    <tr>\n    <td style=\"text-align: left;\">1</td>\n    <td>$ x \\gt \\var{b},$</td>\n    </tr>\n    <tr>\n    <td>&nbsp;</td>\n    <td>&nbsp;</td>\n    </tr>\n    </tbody>\n    </table>\n    <p>c)</p>\n    <p>\\[P(X \\ge \\var{x1})=1-F_X(\\var{x1})=1-\\simplify[std]{({x1}-{a})/{b-a}={b-x1}/{b-a}}\\]</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/"}]}