Could use a little help here,
Let: \begin{align*} \Phi (x) := \int_{-\infty}^x \dfrac{1}{\sqrt{2\pi}}e^{-\dfrac{t²}{2}}\, dt \end{align*}
and I would like to have a result for $x\leq 0$, that it holds: \begin{align*} \Phi (x) \leq e^{-\dfrac{x²}{2}} \end{align*}
Does this hold? If yes, how could one show that?
Wikipedia reference for these formulas.
The cumulative distribution is related to the error function:
$$ \Phi(x) = \frac{1}{2} \operatorname{erfc}\left( -\frac{x}{\sqrt{2}} \right) $$
and the complementary error function satisfies
$$ \operatorname{erfc}\left(x\right) \leq \exp(-x^2) \qquad \qquad x > 0 $$
Combining these gives
$$ \Phi(x) \leq \frac{1}{2} \exp\left( -\frac{x^2}{2} \right) \qquad \qquad x < 0 $$