The general question is if we are given $\mathrm{E} X$ and $\mathrm{Var} X$ for some random variable $X$, can we deduce some information about $|X|$, such as bounds on $\mathrm{E} |X|$ and $\mathrm{Var} |X|$? Note that Chebyshev's and Markov's inequality only provide information about $\mathrm{P} (|X| \geq a)$.
For example, let $X_t, t \geq 0$ be a stochastic process and $a, b, c, d>0$. Suppose we know $ \mathrm{E}(X_t)=-a e^{-b t} \!\ $ and $ \mathrm{Var} (X_t ) = c\left( 1 - e^{-d t} \right). $
Can we deduce some information about $\mathrm{E} |X_t|$ and $\mathrm{Var} |X_t|$, such as bounds on them?
Does $\mathrm{E} |X_t|$ also non-decrease as $t \to \infty$, as $\mathrm{E} X_t$ does? Does $\mathrm{Var} |X_t|$ also non-decrease as $t \to \infty$, as $\mathrm{Var} X_t$ does?
When $X_t$ is normally distributed, $X_t$ is folded normally distributed.Analyzing monotonicity of $\mathrm{E} |X_t|$ and $\mathrm{Var} |X_t|$ is still hard for me to do.
Thanks!