Chapter 3 Discussion Exercise 3 Solution

Question: For a scalar multiplier \(a\), show that \(\mathrm{Var}(ax) = a^2\mathrm{Var}(x)\).

Solution: Using the formula for \(\mathrm{Var}(x)\) from Section 3.1 (using population variance as an example):

\[ \begin{aligned} \mathrm{Var}(ax) &= \frac{1}{n}\sum_{i = 1}^{n}(ax_i - \overline{ax})^2 \\ &= \frac{1}{n}\sum_{i = 1}^{n}(ax_i - a\overline{x})^2 \\ &= \frac{1}{n}\sum_{i = 1}^{n}a^2(x_i - \overline{x})^2 \\ &= a^2\left(\frac{1}{n}\sum_{i = 1}^{n}(x_i - \overline{x})^2\right) \\ &= a^2\mathrm{Var}(x) \end{aligned} \] Back to solutions