Question

If each observation of a dist., whose variance is ${\sigma }^2,$ is multiplied by $\lambda ,$ then the $S.D$. of the new new observations is

• A. $\sigma$
• B. $\lambda \sigma$
• C. $\left|\lambda \right|\sigma$
• D. ${\lambda }^2\sigma$

Answer 1

Answer: (C)

$\left|\lambda \right|\sigma$

We know, $Var(aX+b)=a^2.Var\left(x\right)$
Here $b=0,a=\lambda$
$\therefore$ Required variance $=Var\left(\lambda x\right)={\lambda }^2.Var\left(x\right)={\lambda }^2.{\sigma }^2$
$\Rightarrow S.D\left(\lambda x\right)=|\lambda |.\sigma$