Example 3.2.7.
Recall the setup from Example 3.1.11: a binomial random variable \(S\) is the sum \(H_1 + H_2 + \dotsb + H_n\) of \(n\) indicator random variables, each with parameter \(p\text{.}\) The results of different coin flips are independent from each other, so the random variables \(H_1, \dotsc, H_n\) are independent from each other. From Example 3.2.6, we know that \(\Var(H_i) = p(1-p)\) for each \(i\text{.}\) Therefore:
\begin{align*}
\Var(S) \amp = \Var(H_1 + H_2 + \dotsb + H_n) \\
\amp = \Var(H_1) + \Var(H_2) + \dotsb + \Var(H_n) \\
\amp = \underbrace{p(1-p) + p(1-p) + \dotsb + p(1-p)}_{n \text{ times}} \\
\amp = np(1-p).
\end{align*}
Remember this formula! We’ll make frequent use of it.