// Numbas version: finer_feedback_settings
{"name": "Ricardo's copy of IS2.2", "extensions": ["stats"], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"functions": {}, "ungrouped_variables": ["a", "ux", "xl", "i1", "lo", "j1", "up", "exans", "p", "u", "t", "tol", "ans", "cval", "lx", "xu"], "name": "Ricardo's copy of IS2.2", "tags": ["CFD", "continuous random variables", "cr1", "cumulative distribution functions", "density functions", "distribution function", "distribution functions", "integration", "PDF", "pdf", "probabilities", "probability density function", "random variables", "statistics", "tested1"], "advice": "<h4>a)</h4>\n<p>Note that in order for $f_X(x)$ to be a pdf it must satisfy two important conditions:</p>\n<p>1. $f_X(x) \\ge 0$ in the range $\\var{xl} \\le x \\le \\var{xu}$</p>\n<p>2. The area under the curve given by $f_X(x)$ is $1$ and this implies that:<br/>\\[\\int_{\\var{xl}}^{\\var{xu}}f_X(x)\\;dx = 1\\] as the value of the function is $0$ outside this range.</p>\n<p>We first check condition 2. and then check that condition 1. is satisfied.</p>\n<p>Hence \\[\\begin{eqnarray*} \\int_{-\\infty}^{\\infty}f_X(x)\\;dx=\\int_{\\var{xl}}^{\\var{xu}}f_X(x)\\;dx&amp;=&amp;\\int_{\\var{xl}}^{\\simplify[std]{{xu+xl}/2}}cx\\;dx+\\int_{\\simplify[std]{{xu+xl}/2}}^{\\var{xu}}c(\\var{xu}-x)\\;dx\\\\ &amp;=&amp;\\simplify[std]{{(xu-xl)^2}/8}c+\\simplify[std]{{(xu-xl)^2}/8}c=\\simplify[std]{{(xu-xl)^2}/{4}}c \\end{eqnarray*} \\]<br/>But this has to equal $1$ and so \\[c=\\simplify[std]{4/{p^2}}\\]</p>\n<p>Hence with this value of $c$ we see that condition 2. is satisfied i.e.</p>\n<p>\\[\\int_{-\\infty}^{\\infty}f_X(x)\\;dx=1\\]</p>\n<p>Condition 1. is clearly satisfied as $c \\gt 0$.</p>\n<h4>b)</h4>\n<p>We must have $F_X(x)=0,\\;\\;\\;x \\le \\var{xl},\\;\\;\\;\\textrm{and}\\;\\;\\;F_X(x)=1,\\;\\;\\;x \\ge \\var{xu}$</p>\n<p>Apart from that we find an expression for $F_X(x)$ in each of the ranges $[\\var{xl},\\simplify[std]{{xl+xu}/2}],\\;\\;\\;[\\simplify[std]{{xl+xu}/2},\\var{xu}]$</p>\n<p>1. $x \\in [\\var{xl},\\simplify[std]{{xl+xu}/2}]$</p>\n<p>We have:\\[\\begin{eqnarray*} f_X(x) &amp;=&amp; \\simplify[std]{{4}/{p^2}} x \\\\ \\Rightarrow F_X(x)&amp;=&amp; \\int_{-\\infty}^xf_X(x)\\;dx \\\\ &amp;=&amp; \\simplify[std]{{4}/{p^2}}\\int_{\\var{xl}}^x x\\;dx\\\\ &amp;=&amp;\\simplify[std]{{2}/{p^2}}x^2 \\end{eqnarray*} \\]</p>\n<p>2. $x \\in [\\simplify[std]{{xl+xu}/2},\\var{xu}]$</p>\n<p>We have:\\[\\begin{eqnarray*} f_X(x) &amp;=&amp; \\simplify[std]{{4}/{p^2}}(\\var{xu}-x) \\\\ \\Rightarrow&nbsp; F_X(x)&amp;=&amp; \\int_{-\\infty}^xf_X(x)\\;dx = \\int_{\\var{xl}}^{\\simplify[std]{{xl+xu}/2}}f_X(x)\\;dx+\\int_{\\simplify[std]{{xl+xu}/2}}^x f_X(x)\\;dx\\\\ &amp;=&amp; \\simplify[std]{{4}/{p^2}}\\int_{\\var{xl}}^{\\simplify[std]{{xl+xu}/2}} x\\;dx + \\simplify[std]{{4}/{p^2}}\\int_{\\simplify[std]{{xl+xu}/2}}^x(\\var{xu}-x)\\;dx\\\\ &amp;=&amp;1-\\simplify[std]{{2}/{p^2}}(\\var{xu}-x)^2 \\end{eqnarray*} \\]</p>\n<h4>c)</h4>\n<p>Using the distribution function we have just found we have that:</p>\n<p>$P(\\var{lx} \\lt X \\lt \\var{ux}) = F_X(\\var{ux})-F_X(\\var{lx})$</p>\n<p>But $\\var{ux}$ is in the range between $\\var{(xl+xu)/2}$ and $\\var{xu}$ and the distribution function is given by:</p>\n<p>\\[F_X(x)=1-\\simplify[std]{{2}/{p^2}}(\\var{xu}-x)^2\\]</p>\n<p>Hence $F_X(\\var{ux})=1-\\simplify[std]{{2}/{p^2}}(\\var{xu}-\\var{ux})^2 = \\var{up}$ to 5 decimal places.</p>\n<p>Similarly, $\\var{lx}$ is in the range between $\\var{xl}$ and $\\var{(xl+xu)/2}$ and the distribution function is given by:</p>\n<p>\\[F_X(x)=\\simplify[std]{{2}/{p^2}}x^2\\]</p>\n<p>Hence $F_X(\\var{lx})=\\simplify[std]{{2}/{p^2}}(\\var{lx})^2 = \\var{lo}$ to 5 decimal places.</p>\n<p>Hence $P(\\var{lx} \\lt X \\lt \\var{ux}) = F_X(\\var{ux})-F_X(\\var{lx})=\\var{up}-\\var{lo}=\\var{up-lo}=\\var{ans}$ to 2 decimal places.</p>", "rulesets": {"std": ["all", "fractionNumbers", "!collectNumbers", "!noLeadingMinus"]}, "parts": [{"prompt": "\n      \n      \n      <table>\n      \n      <tr>\n      \n      <td rowspan=\"7\">$f_X(x) = \\left \\{ \\begin{array}{l} \\phantom{{.}} \\\\ \\phantom{{.}} \\\\ \\phantom{{.}} \\\\\\phantom{{.}} \\\\ \\phantom{{.}} \\\\ \\phantom{{.}}\\end{array} \\right .$</td>\n      \n      <td style=\"text-align:left;\">$0$ </td>\n      \n      <td> $ x \\leq \\var{xl},$ </td>\n      \n      </tr>\n      \n      <tr>\n      \n      <td></td>\n      \n      <td></td>\n      \n      </tr>\n      \n      <tr>\n      \n      <td style=\"text-align:left;\">$cx$ </td>\n      \n      <td> $\\var{xl} \\lt x \\leq \\simplify[std]{{xu+xl}/2},$ </td>\n      \n      </tr>\n      \n      <tr>\n      \n      <td></td>\n      \n      <td></td>\n      \n      </tr>\n      \n      <tr>\n      \n      <td style=\"text-align:left;\">$c(\\var{xu}-x)$ </td>\n      \n      <td> $\\simplify[std]{{xu+xl}/2} \\lt x \\leq \\var{xu},$ </td>\n      \n      </tr>\n      \n      <tr>\n      \n      <td></td>\n      \n      <td></td>\n      \n      </tr>\n      \n      <tr>\n      \n      <td style=\"text-align:left;\">$0$</td>\n      \n      <td>$x \\gt \\var{xu}.$</td>\n      \n      </tr>\n      \n      </table>\n      \n      \n      \n      <p>What value of $c$ makes $f_X(x)$ into the pdf of a distribution?</p>\n      \n      \n      \n      <p>Input your answer here as a fraction and not as a decimal.</p>\n      \n      \n      \n      <p>$c=\\;\\;$[[0]]</p>\n      \n      \n      \n        ", "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "gaps": [{"notallowed": {"message": "<p>input as a fraction and not a decimal</p>", "showStrings": false, "strings": ["."], "partialCredit": 0}, "variableReplacements": [], "expectedvariablenames": [], "checkingaccuracy": 0.001, "type": "jme", "showpreview": true, "vsetrangepoints": 5, "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "answersimplification": "std", "scripts": {}, "answer": "{4}/{p^2}", "marks": 2, "checkvariablenames": false, "checkingtype": "absdiff", "vsetrange": [0, 1]}], "showCorrectAnswer": true, "scripts": {}, "marks": 0, "type": "gapfill"}, {"prompt": "\n      \n      \n      <p>Given the value of $c$ found in the first part, determine and input the distribution function $F_X(x)$</p>\n      \n      \n      \n      <table>\n      \n      <tr>\n      \n      <td rowspan=\"7\">$F_X(x) = \\left \\{ \\begin{array}{l} \\phantom{{.}} \\\\ \\phantom{{.}} \\\\ \\phantom{{.}} \\\\ \\phantom{{.}} \\\\ \\phantom{{.}} \\\\ \\phantom{{.}}\\\\ \\phantom{{.}}\\\\ \\phantom{{.}} \\end{array} \\right .$ </td>\n      \n      <td style=\"text-align:left;\">[[0]] </td>\n      \n      <td> $x \\leq \\var{xl},$ </td>\n      \n      </tr>\n      \n      <tr>\n      \n      <td></td>\n      \n      <td></td>\n      \n      </tr>\n      \n      <tr>\n      \n      <td style=\"text-align:left;\">[[1]]</td>\n      \n      <td> $\\var{xl} \\lt x \\leq \\simplify[std]{{xu+xl}/2},$ </td>\n      \n      </tr>\n      \n      <tr>\n      \n      <td></td>\n      \n      <td></td>\n      \n      </tr>\n      \n      <tr>\n      \n      <td style=\"text-align:left;\">[[2]]</td>\n      \n      <td> $\\simplify[std]{{xu+xl}/2} \\lt x \\leq \\var{xu},$ </td>\n      \n      </tr>\n      \n      <tr>\n      \n      <td></td>\n      \n      <td></td>\n      \n      </tr>\n      \n      <tr>\n      \n      <td style=\"text-align:left;\">[[3]]</td>\n      \n      <td> $ x \\gt \\var{xu}.$ </td>\n      \n      </tr>\n      \n      </table>\n      \n      \n      \n      <p>Input all numbers as fractions or integers in the above formulae.</p>\n      \n      \n      \n        ", "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "gaps": [{"vsetrangepoints": 5, "expectedvariablenames": [], "checkingaccuracy": 0.001, "type": "jme", "showpreview": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "scripts": {}, "answer": "0", "marks": 0.5, "checkvariablenames": false, "checkingtype": "absdiff", "vsetrange": [0, 1]}, {"notallowed": {"message": "<p>input numbers as fractions or integers and not as decimals</p>", "showStrings": false, "strings": ["."], "partialCredit": 0}, "variableReplacements": [], "expectedvariablenames": [], "checkingaccuracy": 0.001, "type": "jme", "showpreview": true, "vsetrangepoints": 5, "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "answersimplification": "std", "scripts": {}, "answer": "(({2} / {(p ^ 2)}) * ((x + ( - {xl})) ^ 2))", "marks": 1, "checkvariablenames": false, "checkingtype": "absdiff", "vsetrange": [0, 1]}, {"notallowed": {"message": "<p>input numbers as fractions or integers and not as decimals</p>", "showStrings": false, "strings": ["."], "partialCredit": 0}, "variableReplacements": [], "expectedvariablenames": [], "checkingaccuracy": 0.001, "type": "jme", "showpreview": true, "vsetrangepoints": 5, "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "answersimplification": "std", "scripts": {}, "answer": "(1 + ( - (({2} / {(p ^ 2)}) * ((x + ( - {xu})) ^ 2))))", "marks": 1, "checkvariablenames": false, "checkingtype": "absdiff", "vsetrange": [0, 1]}, {"vsetrangepoints": 5, "expectedvariablenames": [], "checkingaccuracy": 0.001, "type": "jme", "showpreview": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "scripts": {}, "answer": "1", "marks": 0.5, "checkvariablenames": false, "checkingtype": "absdiff", "vsetrange": [0, 1]}], "showCorrectAnswer": true, "scripts": {}, "marks": 0, "type": "gapfill"}, {"prompt": "\n      \n      \n      <p>Also, using the distribution function above find:</p>\n      \n      \n      \n      <p>$P(\\var{lx} \\lt X \\lt \\var{ux})=\\;\\;$[[0]]</p>\n      \n      \n      \n      <p>(input your answer to $2$ decimal places).</p>\n      \n      \n      \n        ", "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "gaps": [{"allowFractions": false, "variableReplacements": [], "maxValue": "ans+tol", "minValue": "ans-tol", "variableReplacementStrategy": "originalfirst", "correctAnswerFraction": false, "showCorrectAnswer": true, "scripts": {}, "marks": 1, "type": "numberentry", "showPrecisionHint": false}], "showCorrectAnswer": true, "scripts": {}, "marks": 0, "type": "gapfill"}], "statement": "<p>A random variable $X$ has a probability density function (<span class=\"caps\">PDF</span>) given by:</p>", "variable_groups": [], "variablesTest": {"maxRuns": 100, "condition": ""}, "preamble": {"css": "", "js": ""}, "variables": {"a": {"definition": "0", "templateType": "anything", "group": "Ungrouped variables", "name": "a", "description": ""}, "ux": {"definition": "j1/2", "templateType": "anything", "group": "Ungrouped variables", "name": "ux", "description": ""}, "xl": {"definition": "0", "templateType": "anything", "group": "Ungrouped variables", "name": "xl", "description": ""}, "i1": {"definition": "round((t+(100-t)*(xu-1))/100)", "templateType": "anything", "group": "Ungrouped variables", "name": "i1", "description": ""}, "lo": {"definition": "precround((2/p^2)*(lx-xl)^2,5)", "templateType": "anything", "group": "Ungrouped variables", "name": "lo", "description": ""}, "j1": {"definition": "round((u*(xu+1)+(100-u)*(2*xu-1))/100)", "templateType": "anything", "group": "Ungrouped variables", "name": "j1", "description": ""}, "up": {"definition": "precround(1-(2/p^2)*(ux-xu)^2,5)", "templateType": "anything", "group": "Ungrouped variables", "name": "up", "description": ""}, "exans": {"definition": "up-lo", "templateType": "anything", "group": "Ungrouped variables", "name": "exans", "description": ""}, "p": {"definition": "(xu-xl)", "templateType": "anything", "group": "Ungrouped variables", "name": "p", "description": ""}, "u": {"definition": "random(0..100)", "templateType": "anything", "group": "Ungrouped variables", "name": "u", "description": ""}, "t": {"definition": "random(0..100)", "templateType": "anything", "group": "Ungrouped variables", "name": "t", "description": ""}, "tol": {"definition": "0.01", "templateType": "anything", "group": "Ungrouped variables", "name": "tol", "description": ""}, "ans": {"definition": "precround(exans,2)", "templateType": "anything", "group": "Ungrouped variables", "name": "ans", "description": ""}, "cval": {"definition": "precround(4/p^2,4)", "templateType": "anything", "group": "Ungrouped variables", "name": "cval", "description": ""}, "lx": {"definition": "i1/2", "templateType": "anything", "group": "Ungrouped variables", "name": "lx", "description": ""}, "xu": {"definition": "xl+random(4,6,8,12,14)", "templateType": "anything", "group": "Ungrouped variables", "name": "xu", "description": ""}}, "metadata": {"notes": "\n        \t\t<p><strong>8/07/2012:</strong></p>\n        \t\t        <p>Added tags.</p>\n        \t\t        <p>Checked calculations, OK.</p>\n        \t\t        <p>Set tolerance via new variable tol=0.01 for last question.</p>\n        \t\t        <p><strong>23/07/2012:</strong></p>\n        \t\t        <p>Added description.</p>\n        \t\t        <p><strong>1/08/2012:</strong></p>\n        \t\t        <p>Added tags.</p>\n        \t\t        <p>In the Advice section, moved \\Rightarrow to the beginning of the line instead of the end of the previous line.</p>\n        \t\t        <p>Question appears to be working correctly.</p>\n        \t\t        <p><strong>21/12/2012:</strong></p>\n        \t\t        <p>Checked calculations, OK. Added tag tested1.</p>\n        \t\t        <p>Checked rounding, OK. Added tag cr1.</p>\n        \t\t", "description": "<p>The random variable $X$ has a PDF which involves a parameter $c$. Find the value of $c$. Find the distribution function $F_X(x)$ and $P(a \\lt X \\lt b)$.</p>", "licence": "Creative Commons Attribution 4.0 International"}, "type": "question", "showQuestionGroupNames": false, "question_groups": [{"name": "", "pickingStrategy": "all-ordered", "pickQuestions": 0, "questions": []}], "contributors": [{"name": "Ricardo Monge", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/471/"}]}]}], "contributors": [{"name": "Ricardo Monge", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/471/"}]}