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    <name>
      <text>38.1.1.6 Cumulative Distribution Function</text>
    </name>
    <questiontext format="html">
      <text><![CDATA[<p class="HELM_exercise">Exercise</p>
<p></p><p>For the pdf in the diagram below<br></p><p style="text-align: center;"><img src="@@PLUGINFILE@@/38.1%20figure%208.svg" alt="Graph of a function where: the line y=0 everywhere except for two line segments - one from (-2,0) to (0,1/2), and the second one from (0,1/2) to (2,0)." width="378" height="159" role="presentation" class="img-responsive atto_image_button_text-bottom"><br></p><p>obtain the cumulative distribution function:</p><p>\( F(x) = \begin{cases} 0 &amp; \text{for all $ x \leq -2$} \\ g(x) &amp; \text{for all $ -2&lt; x \leq 0$} \\ h(x) &amp; \text{for all $ 0 \leq x \leq 2$} \\&nbsp;1 &amp; \text{for all $ x \geq 2$} \end{cases}. \)<br></p><p>\(g(x) = \)&nbsp;[[input:ans1]] [[validation:ans1]] [[feedback:prt1]]</p><p>\(h(x) = \)&nbsp;[[input:ans2]] [[validation:ans2]] [[feedback:prt2]]</p>]]></text>
<file name="38.1 figure 8.svg" path="/" encoding="base64">[encoded data, 11316 characters]</file>
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    <generalfeedback format="html">
      <text><![CDATA[<h4 class="HELM_solution">Solution</h4>
<p>For \(&nbsp;-2&lt; x \leq 0\), we have that</p><p>\[\begin{align*}F(x)&amp;=\displaystyle{\int_{-2}^x} \dfrac{1}{2}+\dfrac{u}{4} \; du\\&amp; = \left.\dfrac{u}{2}\right|_{-2}^x + \left.\dfrac{u^2}{8}\right|_{-2}^x\\&amp; = {@ta1@}.\end{align*}\]</p><p>Hence,&nbsp;<span style="font-size: 0.9375rem;">\(g(x) = \displaystyle{{@ta1@}}.\)</span></p><p>For \(&nbsp;0 \leq x &lt; 2\), we have that<br></p><p>\[\begin{align*}F(x) &amp; = F(0) + \displaystyle{\int_0^x} \dfrac{1}{2}-\dfrac{u}{4} \; du \\&amp; = \dfrac{1}{2} + \left.\dfrac{u}{2}\right|_{0}^x&nbsp; -&nbsp; \left.\dfrac{u^2}{8}\right|_{0}^x\\&amp; ={@ta2@}.\end{align*}\]<br></p><p></p><p>Hence,&nbsp;<span style="font-size: 0.9375rem;">\(h(x) = \displaystyle{{@ta2@}}.\)</span></p><br><p></p>]]></text>
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    <idnumber/>
    <stackversion>
      <text>2021072200</text>
    </stackversion>
    <questionvariables>
      <text>ta1:1/2 +1/2*x + 1/8*x^2;
ta2:1/2 +1/2*x - 1/8*x^2;</text>
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    <questionnote format="moodle_auto_format">
      <text>ONLY ONE VARIANT!</text>
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      <text><![CDATA[<span style="font-size: 1.5em; color:green;"><i class="fa fa-check"></i></span> Correct answer, well done.]]></text>
    </prtcorrect>
    <prtpartiallycorrect format="html">
      <text><![CDATA[<span style="font-size: 1.5em; color:orange;"><i class="fa fa-adjust"></i></span> Your answer is partially correct.]]></text>
    </prtpartiallycorrect>
    <prtincorrect format="html">
      <text><![CDATA[<span style="font-size: 1.5em; color:red;"><i class="fa fa-times"></i></span> Incorrect answer.]]></text>
    </prtincorrect>
    <decimals>.</decimals>
    <scientificnotation>*10</scientificnotation>
    <multiplicationsign>dot</multiplicationsign>
    <sqrtsign>1</sqrtsign>
    <complexno>i</complexno>
    <inversetrig>cos-1</inversetrig>
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