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Question
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
\[\sigma_Y = - \frac{a}{c} \sigma_X\]
\[Y = \frac{aX + b}{c}\]
\[ Y = \frac{\sum^n_{i = 1} \frac{aX + b}{c}}{n}\]
\[ = \frac{\frac{a \sum^n_{i = 1} X + nb}{c}}{n}\]
\[ = \frac{\frac{a}{c} \sum^n_{i = 1} X}{n} + \frac{b}{c}\]
\[ = \frac{aX}{c} + \frac{b}{c}\]
\[\text{ We know: } \]
\[Var (X) = \frac{\sum^n_{i = 1} \left( x_i - X \right)^2}{n}\]
\[ = \sigma^2 \]
\[Var(Y) = \frac{\sum^n_{i = 1} ( y_i - Y )^2}{n}\]
\[ = \frac{\sum^n_{i = 1} \left( \frac{aX}{c} + \frac{b}{c} - \frac{a}{c}X - \frac{b}{c} \right)^2}{n} \]
\[ = \frac{\sum^n_{i = 1} \left( \frac{aX}{c} - \frac{a}{c}X \right)^2}{n}\]
\[ = \left( \frac{a}{c} \right)^2 \frac{\sum^n_{i = 1} \left( x_i - X \right)^2}{n}\]
\[ = \left( \frac{a}{c} \right)^2 \sigma^2 \]
\[SD \text{ of } Y ( \sigma_y ) = \sqrt{\left( \frac{a}{c} \right)^2 \sigma^2}\]
\[ = \left| \frac{a}{c} \right|\sigma\]
\[ac < 0\]
\[ \Rightarrow a < 0 \text{ or } c < 0 \]
\[ \therefore \left| \frac{a}{c} \right| = - \frac{a}{c}\]
\[\Rightarrow \sigma_Y = - \frac{a}{c} \sigma_X\]
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