I am thinking of the property in probability of inequality. In particular, we assume \begin{equation} P[\zeta>a]\leq b, \end{equation} where $a>0$, $b>0$ and $\zeta\in R$ is a random variable.
Now we would like to consider whether the inequality of $P[\zeta^2>a^2]\leq b$ holds.
In fact, for differential and monotonic transformation, e.g., exponential function, the inequality holds. That is, $P[\exp(\zeta)>\exp(a)]\leq b$.
Can someone give hints for me on this issue? Thanks a lot in advance.
$$\zeta = \begin{cases} 1 & w.p. 0.5 \\ -1 & w.p. 0.5\end{cases} $$
Let $a=0.1$, $P(\zeta > 0.1) \leq 0.6$ is a true statements.
However, $P(\zeta^2 > 0.1^2) =1 > 0.6$