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    <name>
      <text>37.1.4.2 three-coin toss</text>
    </name>
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      <text><![CDATA[<p class="HELM_exercise">Exercise</p>
<p>Find the expectation, variance and standard deviation of the number
of heads when tossing {@nCoins@} fair coins.</p>
<strong>Give an exact answer, eg: \(\frac{\sqrt{2}}{2}\)&nbsp;<br>Syntax hint: </strong>\(\sqrt{2}\) can be entered using \(\texttt{sqrt(2)}\).<br><p></p>
<ol class="HELM_parts">
<li>\(E(X) =\) [[input:ans1]] [[validation:ans1]][[feedback:prt1]]
</li><li>\(V(X) =\) [[input:ans2]] [[validation:ans2]][[feedback:prt2]]
</li><li>\(\sigma(X) =\) [[input:ans3]] [[validation:ans3]][[feedback:prt3]]
</li></ol>]]></text>
    </questiontext>
    <generalfeedback format="html">
      <text><![CDATA[<p>The number of heads on \({@nCoins@}\) tosses of a fair coin follows a Binomial distribution with probability function \[f(x) = { {@nCoins@} \choose x } \left(\dfrac{1}{2}\right)^x \left(\dfrac{1}{2}\right)^{{@nCoins@}-x} = { {@nCoins@} \choose x&nbsp;}&nbsp; \dfrac{1}{2^{@nCoins@}}\quad \text{for} \quad x = 0, \ldots, {@nCoins@}.\]</p><ol class="HELM_parts">
<li>The expectation is \(\text{E}(X) = \sum_{x=0}^{@nCoins@} x f(x) =&nbsp; 0\times f(0) + 1\times f(1) + \cdots + {@nCoins@}\times f({@nCoins@}) = {@ta1@}\).</li><li>To calculate the variance, first we compute: 
\[\begin{align*}E(X^2) &amp;= \sum_{x=0}^{@nCoins@} x^2 f(x) \\&amp;=0^2 \times f(0) + 1^2 \times f(1) + \cdots + {@nCoins@}^2 \times f({@nCoins@})\\&amp; = {@EX2@}.\end{align*}\]
It follows that \(\text{V}(X) = E(X^2)-E(X)^2 = {@EX2@} - \left({@ta1@}\right)^2={@ta2@}\).</li><li>Finally, the standard deviation is \(\sigma(X)=\sqrt{V(X)} = \sqrt{{@ta2@}} = {@ta3@}\).<ol></ol></li></ol>]]></text>
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    <stackversion>
      <text>2021072200</text>
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    <questionvariables>
      <text>nCoins: rand([3,4,5]);
EX: sum(i*binomial (nCoins,i)/2^(nCoins),i,0,nCoins);
EX2: sum(i^2*binomial (nCoins,i)/2^(nCoins),i,0,nCoins);
VX: EX2-EX^2;
sX: sqrt(VX);
ta1: EX;
ta2: VX;
ta3: sX;

/*ta1 : 12/8;
ta2 : 3/4;
ta3 : sqrt(3)/2;*/</text>
    </questionvariables>
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      <text><![CDATA[&nbsp;&nbsp;]]></text>
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    <questionnote format="moodle_auto_format">
      <text>\(n: {@nCoins@}, \text{E}(X) = {@ta1@}, \text{E}(X^2) = {@EX2@}, \text{V}(X) = {@ta2@}, \sigma(X) = {@ta3@}\).</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>
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