Given a random variable X and two functions that satisfy: $|f(X)| \leq |g(X)| $ and $Var(f(X)) \neq 0 $ , $Var(g(X)) \neq 0 $.
Is $Var(f(X)) \leq Var(g(x)) $ ?
Trying to prove this I noticed that if $E(f(X)) = E(g(X)) = 0$ then the statement is true but I'm not sure about the general case.
Thank you!
Let $g$ be the constant function $g(x)=100$ and let $$f(x)=\begin{cases} 0 &, x = 0 \\ 1 &, x =1\end{cases}$$
Let $X$ follows Bernoulli($0.5)$., then $Var(f(X)) > 0$ but $Var(g(X))=0$.
Edit since the condition that non-zero variance is added:
We have $Var(f(X))=\frac14$
let $$g(x) = \begin{cases} 2+\epsilon &, x = 0 \\ 2-\epsilon&, x=1 \end{cases}$$
$$Var(g(X))=Var(g(X)-(2-\epsilon))=\frac{4\epsilon^2}{4}=\epsilon^2$$
We can choose $\epsilon$ to be arbitrarily small.