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<article class="definition definition-like"><h4 class="heading">
<span class="type">Definition</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="codenumber">2.2.2</span><span class="period heading-divison-mark heading-divison-mark__period">.</span>
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<div class="para">Let <span class="process-math">\(X\)</span> take values in the interval <span class="process-math">\([a, b]\text{.}\)</span> We say that <span class="process-math">\(X\)</span> has the <dfn class="terminology">uniform distribution</dfn> if, for any subinterval <span class="process-math">\([c, d] \subset [a, b]\text{,}\)</span> the probability that <span class="process-math">\(X\)</span> lies within <span class="process-math">\([c, d]\)</span> is equal to the proportion of the total sample space taken by <span class="process-math">\([c, d]\text{.}\)</span> That is:</div>
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\begin{gather*}
\Pr(c \leq X \leq d) = \frac{d - c}{b - a}.
\end{gather*}
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<div class="para">In this case, <span class="process-math">\(X\)</span> has the pdf <span class="process-math">\(f(x) = \frac{1}{b - a}\text{.}\)</span>
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<div class="autopermalink" data-description="Definition 2.2.2"><a href="#def-continuous-uniform-distribution" title="Copy heading and permalink for Definition 2.2.2" aria-label="Copy heading and permalink for Definition 2.2.2">🔗</a></div></article><span class="incontext"><a class="internal" href="sec-Continuous-RVs.html#def-continuous-uniform-distribution">in-context</a></span>
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