X∼N(0,1), then to prove that for x>0, $$ P(X>x)≤ \frac{1}{2}exp(−x^2/2) $$ I know how to prove the other two kinds of upper-tail inequality for standard normal distribution like this one $$exp(−x^2/2) $$ and this one $$ \frac{exp(−x^2/2)}{x\sqrt{2\pi}} $$ But have no idea how to do this reduced upper-tail inequality. Can anybody help me? Thanks
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How do you get the first bound? – Kavi Rama Murthy Jan 23 '20 at 05:30
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@Kavi Rama Murthy The first bound, exp(-x^2/2), could be obtained by Chernoff bound using moment generating function of standard normal distribution. – Juen Guo Jan 23 '20 at 13:37