If two variates *X* and *Y* are connected by the relation \[Y = \frac{a X + b}{c}\] , where *a*, *b*, *c* are constants such that *ac* < 0, then

#### Options

\[\sigma_Y = \frac{a}{c} \sigma_X\]

\[\sigma_Y = - \frac{a}{c} \sigma_X\]

\[\sigma_Y = \frac{a}{c} \sigma_X + b\]

none of these

#### Solution

\[\sigma_Y = - \frac{a}{c} \sigma_X\]

\[Y = \frac{aX + b}{c}\]

\[ Y = \frac{\sum^n_{i = 1} \frac{aX + b}{c}}{n}\]

\[ = \frac{\frac{a \sum^n_{i = 1} X + nb}{c}}{n}\]

\[ = \frac{\frac{a}{c} \sum^n_{i = 1} X}{n} + \frac{b}{c}\]

\[ = \frac{aX}{c} + \frac{b}{c}\]

\[\text{ We know: } \]

\[Var (X) = \frac{\sum^n_{i = 1} \left( x_i - X \right)^2}{n}\]

\[ = \sigma^2 \]

\[Var(Y) = \frac{\sum^n_{i = 1} ( y_i - Y )^2}{n}\]

\[ = \frac{\sum^n_{i = 1} \left( \frac{aX}{c} + \frac{b}{c} - \frac{a}{c}X - \frac{b}{c} \right)^2}{n} \]

\[ = \frac{\sum^n_{i = 1} \left( \frac{aX}{c} - \frac{a}{c}X \right)^2}{n}\]

\[ = \left( \frac{a}{c} \right)^2 \frac{\sum^n_{i = 1} \left( x_i - X \right)^2}{n}\]

\[ = \left( \frac{a}{c} \right)^2 \sigma^2 \]

\[SD \text{ of } Y ( \sigma_y ) = \sqrt{\left( \frac{a}{c} \right)^2 \sigma^2}\]

\[ = \left| \frac{a}{c} \right|\sigma\]

\[ac < 0\]

\[ \Rightarrow a < 0 \text{ or } c < 0 \]

\[ \therefore \left| \frac{a}{c} \right| = - \frac{a}{c}\]

\[\Rightarrow \sigma_Y = - \frac{a}{c} \sigma_X\]