Question

If X and Y are two variates connected by the relation

\[Y = \frac{aX + b}{c}\]  and Var (X) = σ², then write the expression for the standard deviation of Y.

 

 

Answer 1

\[Y = \frac{aX + b}{c}\]
\[Y = \frac{\sum y_i}{n} = \frac{\frac{a\sum X + nb}{c}}{n}\]
\[ = \frac{a\sum X}{nc} + \frac{nb}{nc}\]
\[ = \frac{a \bar{X}}{c} + \frac{b}{c}\]

We know:

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

\[ = \sigma^2 \]

\[Var\left( Y \right) = \frac{\sum \left( y_i - \bar{Y} \right)^2}{n}\]

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

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

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

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

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

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