Question

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

• A. ${\sigma }_Y=\frac{a}{c}{\sigma }_X$
• B. ${\sigma }_Y=-\frac{a}{c}{\sigma }_X$
• C. ${\sigma }_Y=\frac{a}{c}{\sigma }_X+b$
• D. none of these

Answer 1

The correct option is B ${\sigma }_Y=-\frac{a}{c}{\sigma }_X$

For $\left(\frac{aX+b}{c}\right)$
${\sigma }_Y=\sigma \left(\frac{aX+b}{c}\right)=\left|\frac{a}{c}\right|{\sigma }_X$
and ac<0 therefore a and c are of opposite sign.
so, $\frac{a}{c}<0$
therefore $\left|\frac{a}{c}\right|=-\frac{a}{c}$
${\sigma }_Y=-\frac{a}{c}{\sigma }_X$