Question

If the standard deviation of $X$ is $\sigma$, then s.d. of the variable $U=\frac{aX+b}{c}$ where $a,b,c$ are constants is

• A. $\left|\frac{c}{a}\right|\sigma$
• B. $\left|\frac{a}{c}\right|\sigma$
• C. $\left|\frac{b}{c}\right|\sigma$
• D. $\frac{c^2}{a^2}\sigma$

Answer 1

The correct option is B $\left|\frac{a}{c}\right|\sigma$

S.D of $X=\sigma$
$U=\frac{aX+b}{c}X\to x_1,x_2,...x_{n}U\to \frac{a{x}_1+b}{c},\frac{a{x}_2+b}{c},...\frac{x_n+b}{c}$
We know that
$Var(pX+q)=p^{2}Var\left(X\right)$
So, $Var\left(U\right)=Var\left(\frac{aX+b}{c}\right)=Var(\frac{aX}{c}+\frac{b}{c})={\left(\frac{a}{c}\right)}^{2}Var\left(X\right)$
So, S.D of $U=\sqrt{{\left(\frac{a}{c}\right)}^{2}Var\left(X\right)}=\left|\frac{a}{c}\right|\sigma$