Question

If x₁, x₂, ..., x_n are n values of a variable X and y₁, y₂, ..., y_n are n values of variable Y such that y_i = ax_i + b; i = 1, 2, ..., n, then write Var(Y) in terms of Var(X).

Answer 1

$$\begin{gathered} Var\left(X\right)=\frac{\sum {\left(x_i-\overline{X}\right)}^2}{n} \\ Var\left(Y\right)=\frac{\sum {\left(y_i-\overline{Y}\right)}^2}{n} \end{gathered}$$

We have:
$$\begin{gathered} y_i=a{x}_i+b \\ \overline{y}=\frac{\sum y_i}{n}=\frac{a\sum x_i+nb}{n}=a\overline{X}+b \end{gathered}$$
$\mathrm{Var}\left(\mathrm{Y}\right)=\frac{{\sum }_{}{\left(a{x}_i+b-a\overline{X}-b\right)}^2}{n}=\frac{\sum {\left(a{x}_i-a\overline{X}\right)}^2}{n}=a^2\frac{\sum {\left(x_i-\overline{X}\right)}^2}{n}=a^{2}Var\left(X\right)$