Example 3.1.11.
Suppose \(S \sim \Bin(n, p)\text{.}\) Definition 2.1.6 describes a binomially distributed random variable as counting the number of occurrences of some event, which either happens or not in each of \(n\) independent trials. This leads us to a very natural idea: identify each individual possible occurrence of the event, and assign it an indicator random variable. In this case, let \(H_i\) indicate that flip \(i\) comes up heads. Then
\begin{gather*}
S = H_1 + H_2 + \dotsb + H_n.
\end{gather*}
We already know (Example 3.1.5) that the expected value of each \(H_i\) is the probability of the indicated event—that flip \(i\) is heads. This is precisely the parameter \(p\text{.}\) Therefore:
\begin{align*}
\E(S) \amp = \E(H_1 + H_2 + \dotsb + H_n) \\
\amp = \E(H_1) + \E(H_2) + \dotsb + \E(H_n) \\
\amp = \underbrace{p + p + \dotsb + p}_{n \text{ times}} \\
\amp = np.
\end{align*}
Remember this formula! We’ll make frequent use of it.