Let $X : \Omega \rightarrow \mathbb{R}$ be a real-valued random variable. We assume that there exist two strictly positive reals $a$ and $b$ such that, for all non-negative reals $t$,
$$\mathbb{P}(|X| \geqslant t) \leqslant a \exp(-bt^{2})$$
Show that the second moment of $X$ is less than or equal to $\frac{a}{b}$.