$$\text { If random variables } x _ { 1 } \text { and } x _ { 2 } \text { are independent, then } \sigma _ { x _ { 1 } + x _ { 2 } } ^ { 2 } = \sigma _ { x _ { 1 } } ^ { 2 } + \sigma _ { x _ { 2 } } ^ { 2 }$$

Question

Quizplus · Accepted Answer

If two random variables, $x_1$ and $x_2$, are independent, their variances add up. That is, $\operatorname{Var}(x_1+x_2) = \operatorname{Var}(x_1)+\operatorname{Var}(x_2)$. Taking the square root of both sides gives the desired result. That is, $\sigma_{x_1+x_2} = \sqrt{\operatorname{Var}(x_1+x_2)} = \sqrt{\operatorname{Var}(x_1)+\operatorname{Var}(x_2)} = \sqrt{\sigma_{x_1}^2+\sigma_{x_2}^2}$.