$\def\Var{\operatorname{Var}}\Var \left[\frac{(n-1)S^2}{\sigma^2} \right] = 2(n-1) \Longrightarrow \Var(S^2) = \frac{2\sigma^4}{(n-1)}$

Question

$\def\Var{\operatorname{Var}}$ $$\Var \left[\frac{(n-1)S^2}{\sigma^2} \right] = 2(n-1) \Longrightarrow \Var(S^2) = \frac{2\sigma^4}{(n-1)}$$

I know that when you take a constant out of the variance, you square it... but this implies that $\sigma^2$ was squared too — why do we treat $\sigma$ as a constant?

Answer 1

Hint:

$$ \sigma^{2}=E(X^{2})-(EX)^{2} $$ is a number because expectation of a random variable is a number. Try $E(EX)=EX$.