Let x1, x2, ..., xn be values taken by a variable X and y1, y2, ..., yn be the values taken by a variable Y such that yi = axi + b, i = 1, 2,..., n. Then,
Options
Var (Y) = a2 Var (X)
Var (X) = a2 Var (Y)
Var (X) = Var (X) + b
none of these
Solution
Var (Y) = a2 Var (X)
\[\text{ Var } (X) = \frac{\sum^n_{i = 1} ( x_i - \bar{X}^{} )^2}{n} \text{ where Mean } \left( X \right) = \frac{\sum^n_{i = 1} x_i}{n}\]
\[\text{ Var } (Y) = \frac{\sum\nolimits_{i = 1}^n ( y_i - Y )^2}{n} \text{ and } Y = \frac{\sum^n_{i = 1} y_i}{n}\]
\[\text{ We have } , \]
\[ y_i = a x_i + b\]
\[Y = \frac{\sum\nolimits_{i = 1}^n y_i}{n}\]
\[ = \frac{\sum\nolimits_{i = 1}^n a x_i + b}{n}\]
\[ = \frac{a \sum\nolimits_{i = 1}^n x_i}{n} + \frac{nb}{n} \]
\[ = aX + b\]
\[\text{ Var } (Y) = \frac{\sum^n_{i = 1} \left( y_i - Y \right)^2}{n}\]
\[ = \frac{\sum^n_{i = 1} \left\{ a x_i + b - \left( aX + b \right) \right\}^2}{n}\]
\[ = \frac{\sum^n_{i = 1} (a x_i - aX )^2}{n}\]
\[ = a^2 \frac{\sum^n_{i = 1} ( x_i - X )^2}{n}\]
\[ = a^2 \text{ Var } (X)\]
