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 $t$, $$\mathbb{P}(|X| \geqslant t) \leqslant a \exp(-bt^{2})$$ Let $\delta$ be a real such that $0 \leqslant |\delta| \leqslant \sqrt{\frac{a}{b}}$. Justify that the inequality $$\mathbb{P}(|X + \delta| \geqslant t) \leqslant a \exp(a) \exp\left(-\frac{1}{2}bt^{2}\right)$$ remains valid if $0 \leqslant t < |\delta|$.
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 $t$,
$$\mathbb{P}(|X| \geqslant t) \leqslant a \exp(-bt^{2})$$
Let $\delta$ be a real such that $0 \leqslant |\delta| \leqslant \sqrt{\frac{a}{b}}$. Justify that the inequality
$$\mathbb{P}(|X + \delta| \geqslant t) \leqslant a \exp(a) \exp\left(-\frac{1}{2}bt^{2}\right)$$
remains valid if $0 \leqslant t < |\delta|$.