Let $Y \sim \mathcal{N}(0,1)$ and $\epsilon > 0$. I would like to prove the following $$\mathbb{P}(Y \geq \sqrt{2 \ln{\frac{2}{\epsilon}}} - \frac{1}{2\sqrt{2 \ln{\frac{2}{\epsilon}}}}) \leq \epsilon.$$

Any tips would be greatly appreciated!

  • $\begingroup$ Have you already tried some Chebyshev techniques ? $\endgroup$ – ippiki-ookami Jul 18 '18 at 14:30
  • $\begingroup$ I did, however was unable to come up with the proof using Chebyshev inequality. $\endgroup$ – Metod Jazbec Jul 18 '18 at 14:55
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    $\begingroup$ In this post there is an iterative method with integration by parts to compute upper and lower bounds for the tail probabilities. $\endgroup$ – ippiki-ookami Jul 19 '18 at 7:06
  • $\begingroup$ Thanks a lot, this is exactly what I was looking for! $\endgroup$ – Metod Jazbec Jul 19 '18 at 7:55

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