A problem of proving that a certain concentration inequality cannot be improved

Consider the following concentration inequality.

The problem is :

The inequalities are easily proven with the help of Markov's inequality. But I cannot find a strategy to construct an example to show that it cannot be improved. My rough strategy is : For every $\epsilon>0$, to construct a random variable $X_{\epsilon}$ with $E(X_{\epsilon})=0$ and $var(X_{\epsilon})=\sigma^2$ such that $P(X_{\epsilon} \geq x)>\frac{\sigma^2}{\sigma^2+x^2}$ if $x>0$ and $P(X_{\epsilon} \geq x)<\frac{x^2}{\sigma^2+x^2}$ if $x<0$. How can I construct such $X_{\epsilon}$? Thanks in advance.

Source : Rohatgi-Saleh, p.$98$, problem $3$.

Why not let $X\equiv 0$? That achieves equality in both cases.

• Can I imply that the CI's cannot be improved upon (for any random variable satisfying the prescribed conditions) just by producing an example where equality is achieved? Commented Mar 4, 2018 at 3:13
• @Samayita That's correct. If you produce an example where equality is achieved, you can't possibly produce a lower or higher bound otherwise it won't work for that example.
– user223391
Commented Mar 4, 2018 at 3:23
• For example we still may have bounds which are sharper or equal to the bounds given in the lemma, and both the bounds coincide (and achieves equality) in the particular example you've given Commented Mar 4, 2018 at 3:24
• @Samayita Yes, other random variables may indeed be lower or higher than the bounds we have.
– user223391
Commented Mar 4, 2018 at 3:25
• Okay I understand now, the point is whether or not the bounds can be improved for ALL random variables (satisfying the prescribed conditions). Thanks! Commented Mar 4, 2018 at 3:31