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      <text>37.1.2.3 Example turbo generator</text>
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<h4 class="HELM_example">Example</h4>
<p>Turbo Generators plc manufacture seven large turbines for a customer. Three of these turbines do not meet the customer’s specification. Quality control inspectors choose two turbines at random. Let the discrete random variable <span class="math inline">\(X\)</span> be defined to be the number of turbines inspected which meet the customer’s specification.</p>
<ol class="HELM_parts">
<li>Find the probabilities that <span class="math inline">\(X\)</span> takes the values <span class="math inline">\(0, 1\)</span> or <span class="math inline">\(2\)</span>
</li><li> Find and graph the cumulative distribution function.
</li></ol>

<h4 class="HELM_solution">Solution</h4>
<ol class="HELM_parts">
<li> <p>The possible values of <span class="math inline">\(X\)</span> are clearly <span class="math inline">\(0, 1\)</span> or <span class="math inline">\(2\)</span> and may occur as follows: <span class="math inline">\(\qquad\begin{matrix}
\text{Sample Space}\qquad\qquad&amp;\text{Value of}\ X\\
\text{Turbine faulty, Turbine faulty}\hfill&amp;0\\
\text{Turbine faulty, Turbine good}\hfill&amp;1\\
\text{Turbine good, Turbine faulty}\hfill&amp;1\\
\text{Turbine good, Turbine good}\hfill&amp;2
\end{matrix}\)</span> <span class="math inline">\(\phantom{(a)\ }\)</span>
<br>We can easily calculate the probability that <span class="math inline">\(X\)</span> takes the values <span class="math inline">\(0,1\)</span> or <span class="math inline">\(2\)</span> as follows: 
<br>\(\text{P}(X=0)=\dfrac{3}{7}\times\dfrac{2}{6}=\dfrac{1}{7}\)
<br>\(\text{P}(X=1)=\dfrac{4}{7}\times\dfrac{3}{6}+\dfrac{3}{7}\times\dfrac{4}{6}=\dfrac{4}{7}\)
<br>\(\text{P}(X=2)=\dfrac{4}{7}\times\dfrac{3}{6}=\dfrac{2}{7}\)
<br>The values of <span class="math inline">\(\displaystyle{F(x)=\sum_{x_i\le x}\text{P}(X=x_i)}\)</span> are clearly <span class="math inline">\(F(0)=\dfrac{1}{7}\qquad F(1)=\frac{5}{7}\qquad\text{and}\qquad
F(2)=\dfrac{7}{7}=1\)</span></p>
</li><li> The graph of the step function \(F(x)\) is shown: <img src="@@PLUGINFILE@@/37-1-fig1-stepfunction.svg" alt="Graph of cumulative distribution function with F(0) = 1/7, F(1) = 5/7, F(2) = 1" class="img-responsive atto_image_button_text-bottom" width="437" height="219"><br></li></ol>]]></text>
<file name="37-1-fig1-stepfunction.svg" path="/" encoding="base64">[encoded data, 9488 characters]</file>
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