Compare the variances of $X^2$, $X$ and $\sqrt{X}$.

Question

Compare the variances of $X^2$, $X$ and $\sqrt{X}$.

Is there a way to effectively compare sizes using the variance equation below?

$$Var()=[(−[])^2] = [^2]−([])^2$$

Answer 1

How they compare will depend upon the distribution. Let's focus on the variances of $X^2$ and $X$. Suppose, for instance,

$$X=B+a,B\sim Bern(p),a\in\mathbb{R}$$

so $$X^2=(1+2a)B+a^2.$$

Then $$\text{Var}(X^2)=(1+2a)^2\text{Var}(X),$$

which is less than $\text{Var}(X)$ for $a\in (-1,0)$ but is otherwise at least as great as $\text{Var}(X).$