Example 3.1.2.
If \(X\) takes on the values \(x_1, x_2, \dotsc, x_n\) uniformly (i.e., each value having probability \(\frac{1}{n}\)), then the expected value is the usual average:
\begin{align*}
\E(X) \amp = x_1 \cdot \frac{1}{n} + x_2\cdot \frac{1}{n} + \dotsb + x_n \cdot \frac{1}{n} \\
\amp = \frac{x_1 + x_2 + \dotsb + x_n}{n}
\end{align*}
For example, the expected value of a fair 6-sided die roll \(R\) is:
\begin{align*}
\E(R) \amp = 1 \left(\frac{1}{6}\right) + 2 \left(\frac{1}{6}\right) + 3 \left(\frac{1}{6}\right) + 4 \left(\frac{1}{6}\right) + 5 \left(\frac{1}{6}\right) + 6 \left(\frac{1}{6}\right) \\
\amp = \frac{1 + 2 + \dotsb + 6}{6} = \frac{21}{6} = \frac{7}{2}.
\end{align*}