Exercise C.3.9

Show that for any random variable $X$ that takes on only the values $0$ and $1$, we have $\Var[X] = \E[X]\E[1-X]$.

Let's first calculate the expectations:

$$ \E[X] = 0 \cdot \Pr\{X = 0\} + 1 \cdot \Pr\{X = 1\} = \Pr\{X = 1\} \\ \E[1-X] = \Pr\{X = 0\} \\ \E[X]\E[1-X] = \Pr\{X = 0\} \cdot \Pr\{X = 1\} $$

Now - the variance:

$$ \Var[X] = \E[X^2] - \E^2[X] = \Pr\{X = 1\} - (\Pr\{X = 1\})^2 = \Pr\{X = 1\} (1 - \Pr\{X = 1\}) = \Pr\{X = 0\} \cdot Pr\{X = 1\} $$