<p>A continuous random variable is a theoretical representation of a continuous variable such as height, mass or time.</p>
<p>A <strong>continuous random variable</strong> is specified by its probability density function (<strong>p.d.f.</strong>) written as&nbsp;\(\mathrm{f}(x)\).</p>
<p>If&nbsp;\(X\) is a continuous random variable with probability density function \(\mathrm{f}(x)\) valid over the range \(a \le x \le b\), then</p>
<ol class="i-list"><li><div class="list-item-content"><span class="list-item-text">\(\mathrm{f}(x) \ge 0\)</span></div></li><li><div class="list-item-paragraph">\(\int_{\text{all}\hspace{0.5em} x} \mathrm{f}(x) \, \mathrm{d}x = 1\), i.e,<br>\[
\int_{a}^{b} \mathrm{f}(x) \, \mathrm{d}x = 1.
\]</div></li></ol>
<p></p>
<ol class="i-list" start="3"><li><div class="list-item-content"><span class="list-item-text">\(\mathrm{P}(X \le x_1) = \int_{a}^{x_1} \mathrm{f}(x) \, \mathrm{d}x\), where&nbsp;\(a \le x_1 \le x_2 \le b\).</span></div></li><li><div class="list-item-content"><span class="list-item-text">\(\mathrm{P}(X \ge x_2) = \int_{x_2}^{b} \mathrm{f}(x) \, \mathrm{d}x\), where&nbsp;\(a \le x_1 \le x_2 \le b\).</span></div></li><li><div class="list-item-content"><span class="list-item-text">\(\mathrm{P}(x_1 \le X \le x_2) = \int_{x_1}^{x_2} \mathrm{f}(x) \, \mathrm{d}x\), where&nbsp;\(a \le x_1 \le x_2 \le b\).</span></div></li></ol>
<div class="content-block"><figure class="question-figure aligncenter" style="text-align: center;"><img src="https://vocwkpgapvksartxhhor.supabase.co/storage/v1/object/public/question-images/pasted/1777623493973-znqz0r2lfc.png" style="max-width: min(400px, 100%); height: auto; display: block;" class="aligncenter custom-width" /></figure></div>
<p><strong>Note:</strong></p>
<p>\[
\begin{align*}
\mathrm{P}(x_1 \le X \le x_2) 
& = \mathrm{P}(X = x_1) + \mathrm{P}(x_1 &lt; X &lt; x_2) + \mathrm{P}(X = x_2) \\
& = 0 + \mathrm{P}(x_1 &lt; X &lt; x_2) + 0  \\
& = \mathrm{P}(x_1 &lt; X &lt; x_2).
\end{align*}
\]</p>
<p>\[\therefore \mathrm{P}(x_1 \le X \le x_2) = \mathrm{P}(x_1 &lt; X &lt; x_2) = \mathrm{P}(x_1 \le X &lt; x_2) = \mathrm{P}(x_1 &lt; X \le x_2).\]</p>

A continuous random variable is a theoretical representation of a continuous variable such as height, mass or time.

A **continuous random variable** is specified by its probability density function (**p.d.f.**) written as \(\mathrm{f}(x)\).

If \(X\) is a continuous random variable with probability density function \(\mathrm{f}(x)\) valid over the range \(a \le x \le b\), then

\(\int_{\text{all } x} \mathrm{f}(x) \, \mathrm{d}x = 1\), i.e,
\[
\int_{a}^{b} \mathrm{f}(x) \, \mathrm{d}x = 1.
\]

\(\mathrm{P}(X \le x_1) = \int_{a}^{x_1} \mathrm{f}(x) \, \mathrm{d}x\), where \(a \le x_1 \le x_2 \le b\).

\(\mathrm{P}(X \ge x_2) = \int_{x_2}^{b} \mathrm{f}(x) \, \mathrm{d}x\), where \(a \le x_1 \le x_2 \le b\).

\(\mathrm{P}(x_1 \le X \le x_2) = \int_{x_1}^{x_2} \mathrm{f}(x) \, \mathrm{d}x\), where \(a \le x_1 \le x_2 \le b\).

\[
\begin{align*}
\mathrm{P}(x_1 \le X \le x_2) 
& = \mathrm{P}(X = x_1) + \mathrm{P}(x_1 < X < x_2) + \mathrm{P}(X = x_2) \\
& = 0 + \mathrm{P}(x_1 < X < x_2) + 0  \\
& = \mathrm{P}(x_1 < X < x_2).
\end{align*}
\]

\[\therefore \mathrm{P}(x_1 \le X \le x_2) = \mathrm{P}(x_1 < X < x_2) = \mathrm{P}(x_1 \le X < x_2) = \mathrm{P}(x_1 < X \le x_2).\]

Probability density or distribution function

Probability density or distribution function