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      <text>39.1.2.1 The Standard Normal Distribution</text>
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      <text><![CDATA[<h3>The Standard Normal Distribution</h3>

<p>At this stage we shall, for simplicity, consider what is known as a standard normal distribution which is obtained by choosing particularly simple values for \(\mu\) and \(\sigma\).</p>

<div class="HELM_keypoint">
<h4>Key Point</h4>
<p>The <b>standard normal distribution</b> has a mean of <b>zero</b> and a variance of <b>one</b></p>
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<p>In the Figure below, we show the graph of the standard normal distribution which has probability density function \(y=\dfrac{1}{\sqrt{2\pi}}\ e^{-x^2/2}\).</p> 

<p></p><center><img src="@@PLUGINFILE@@/39-1-fig-2-standardnormaldist.svg" alt="The Standard Normal Distribution Curve" width="332" height="87" class="img-responsive atto_image_button_text-bottom"><br></center><p></p> 
<p></p><center><b>Figure</b>: The standard normal distribution curve</center><p></p> 

<p>The result which makes the standard normal distribution so important is as follows:</p> 

<div class="HELM_keypoint">
<h4>Key Point</h4>
<p>If the behaviour of a continuous random variable X is described by the distribution \(N(\mu, \sigma^2)\), then the behaviour of the random variable \(Z = \dfrac{X-\mu}{\sigma}\) is described by the standard normal distribution \(N(0, 1)\).</p>
<p>We call \(Z\) the standardised normal variable and we write \(Z ∼ N(0, 1)\)</p>
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<hr>
<h4 class="HELM_example">Example</h4>
<p>If the random variable \(X\) is described by the distribution \(N(45,0.000625)\), then what is the transformation required to obtain the standardised normal variable?</p> <p>
</p><h4 class="HELM_solution">Solution</h4>
<p> Here, \(\mu = 45\) and \(\sigma^2 = 0.000625\), so that \(\sigma = 0.025\). Hence \(Z = \dfrac{X-45}{0.025}\) is the required transformation.</p>
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<h4 class="HELM_example">Example</h4>
<p>When the random variable \(X\sim N(45,0.000625)\) takes values between \(44.95\) and \(45.05\), between which values does the random variable \(Z\) lie?</p> 
<h4 class="HELM_solution">Solution</h4>
<p>When \(X = 45.05, Z = \dfrac{45.05 − 45}{0.025} = 2\)</p>
<p>When \(X = 44.95, Z = \dfrac{44.95 − 45}{0.025} = -2\)</p>
<p>Hence \(Z\) lies between \(−2\) and \(2\).</p>]]></text>
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