#### Definition 2.2.2.

Let \(X\) take values in the interval \([a, b]\text{.}\) We say that \(X\) has the uniform distribution if, for any subinterval \([c, d] \subset [a, b]\text{,}\) the probability that \(X\) lies within \([c, d]\) is equal to the proportion of the total sample space taken by \([c, d]\text{.}\) That is:
\begin{gather*} \Pr(c \leq X \leq d) = \frac{d - c}{b - a}. \end{gather*}
In this case, \(X\) has the pdf \(f(x) = \frac{1}{b - a}\text{.}\)
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