Is $\int_x^{\infty}e^{-\frac{t^2}{2}} < \frac{1}{x}e^{-\frac{x^2}{2}}$?

Question

While solving a problem in real analysis, I got stuck. I need to prove $$\int_x^{\infty}e^{-\frac{t^2}{2}}dt < \frac{1}{x}e^{-\frac{x^2}{2}} $$

Clearly I have to use some kind of inequality, but cant figure out how to proceed further.

Thanks for the help.

Answer 1

Use

$$e^{-t^2/2} = \frac{t}{t}e^{-t^2/2} < \frac{t}{x}e^{-t^2/2}$$

for $t > x$.

Answer 2

This is (almost) the Gaussian Tail inequality. The proof can be found on this page.

The rest of contains many similar inequalities which can also be of interest. For example, you could also use Markov's inequality to prove it in a line or two.