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So we have the jensen's inequality: $$|EX| \leq E|X|$$

Any bound on the Jensen gap (upper bound or lower bound)? $$\text{gap}=E|X| - |EX|$$

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    $\begingroup$ I think it is off-topic to ask "Here is a quantity, what are bounds on it?", like you've done here. You need to be more specific; what is the bound allowed to be in terms of? Ideally, what do you need the bound for? The fact that you have received two factually correct answers and are satisfied with neither further suggests that your problem is under-specified. $\endgroup$ Sep 30, 2021 at 18:39

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The gap can be arbitrarily large. For instance, if $X$ is a random variable so that $X(0) = -N$ and $X(1)=N$, and the events $0$ and $1$ have probability $1/2$, then $|E(X)| = |\frac{1}{2}N - \frac{1}{2}N|=0$, but $E(|X|) = N$.

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  • $\begingroup$ Is this bound have any mathematical property? $\endgroup$ Sep 30, 2021 at 18:06
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    $\begingroup$ Not sure what you are asking for. $\endgroup$ Sep 30, 2021 at 18:09
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$0 \le |EX| \leq E|X|$ so $$\text{gap} \le E|X|.$$ Reijo provided an example where $\text{gap} = E|X|$.

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  • $\begingroup$ Thanks GEdgar, is there anymore mathematical property that we can say on this particular gap, for instance, I saw it somewhere that the gap is normally distributed or something $\endgroup$ Sep 30, 2021 at 18:18
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    $\begingroup$ @NygenPatricia The gap is a number, there is nothing random about it. It doesn't have a distribution, unless you want to consider it a distribution that puts all of the probability on a single point. $\endgroup$ Sep 30, 2021 at 18:50

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