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      <text>38.1.2.1 Mean and Variance of a Continuous Distribution</text>
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      <text><![CDATA[<h5>Mean and Variance of a Continuous Distribution</h5>
<p>You will probably have realised by now that, essentially, the definitions of discrete and continuous
random variables are virtually the same provided we use the analogues given in the following table:<br></p><p><br>
</p><table style="text-align: center; width: 90%;">
<caption></caption>
<thead>
<tr>
<th scope="col">Quantity</th>
<th scope="col">Discrete Variable</th>
<th scope="col">Continuous Variable</th>
</tr>
</thead>
<tbody>
<tr>
<td>Probability</td>
<td>\(\text{P}(X=x)\)</td>
<td>\(f(x) \; dx\)</td>
</tr>
<tr>
<td>Allowed Values</td>
<td>\(\text{P}(X =x) \geq 0 \)</td>
<td>\(f(x) \geq 0\)</td>
</tr>
<tr>
<td>Summation</td>
<td>\(\displaystyle{\sum \text{P}(X=x)} \)</td>
<td>\(\displaystyle{\int} f(x) \; dx\)</td>
</tr>
<tr>
<td>Expectation</td>
<td>\(\text{E}(X)= \displaystyle{\sum x\text{P}(X=x)}\)</td>
<td>?</td>
</tr>
<tr>
<td>Variance</td>
<td>\(\text{V}(X) = \displaystyle{\sum(x-\text{E}(X))^2\text{P}(X=x)}\)</td>
<td>?</td>
</tr>
</tbody>
</table>
<p style="text-align: center;"><br></p><p>Completing the above table of analogues to write down the mean and variance of a continuous
variable leads to the obvious definitions given in the following Key Point:</p>
<div class="HELM_keypoint">
<h4>Key Point&nbsp;</h4>
<p>Let \(X\) be a continuous random variable with associated pdf \(f(x)\). Then its expectation and variance
denoted by \(E[X]\) (or \(\mu\)) and \(V[X]\) (or \(\sigma^2\) ) respectively are given by: 
\[\text{E}(X)=\displaystyle{\int}_{-\infty}^{\infty} x f(x) \; dx\] and 
\[\text{V}(X) =\displaystyle{\int}_{-\infty}^{\infty} (x-\text{E}(X))^2 f(x) \; dx =&nbsp;\int_{-\infty}^{\infty} x^2f(x) \; dx -\text{E}(X)^2. \]</p></div><br>
<p>As with discrete random variables the variance \(V[X]\) can be written in an alternative form, more
amenable to calculation:\[\text{V}(X) = \text{E}(X^2)-\text{E}(X)^2\] where
\[\text{E}(X^2)= \displaystyle{\int}_{-\infty}^{\infty} x^2f(x) \; dx.\]</p>]]></text>
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