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!
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Have you already tried some Chebyshev techniques ? – ippiki-ookami Jul 18 '18 at 14:30
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I did, however was unable to come up with the proof using Chebyshev inequality. – Metod Jazbec Jul 18 '18 at 14:55
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1In this post there is an iterative method with integration by parts to compute upper and lower bounds for the tail probabilities. – ippiki-ookami Jul 19 '18 at 07:06
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Thanks a lot, this is exactly what I was looking for! – Metod Jazbec Jul 19 '18 at 07:55