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
$$\mathbb{E}(X^{2}) = 2 \int_{0}^{+\infty} t \mathbb{P}(|X| \geqslant t) \, dt$$
You may denote $X^{2}(\Omega) = \{y_{1}, \ldots, y_{n}\}$ with $0 \leqslant y_{1} < y_{2} < \cdots < y_{n}$.