Sharp two-sided Gaussian tail bound
= Sharp two-sided Gaussian tail bound
{title2=$\mathbb P(|G|>t)\leq e^{-t^2/2}$}
For $G$ with the <standard normal distribution> and $t\geq0$, $\mathbb P(|G|>t)=2(1-\Phi(t))\leq e^{-t^2/2}$. The difference has derivative $e^{-t^2/2}(\sqrt{2/\pi}-t)$, starts at zero, and tends to zero, proving its nonnegativity.