probability theory proof of exponential chebyshev inequality 
This is a question about my homework.
I am not sure about what is exponential Chebyshev inequality, also how do I get rid of the absolute value and prove it directly by PDF?
As well as the exponential Chebyshev bound.
Thank you

By symmetry of the Normal density about $0$
\begin{align}
      P[|Z|>a]
&=    2P[Z>a] \\
&=    2 \int_a^\infty f(x) dx \\
&=    \frac{\sqrt{2}}{\sqrt{\pi}}\int_a^\infty \exp(-x^2/2) dx \\
&\leq \frac{\sqrt{2}}{a\sqrt{\pi}}\int_a^\infty x\exp(-x^2/2) dx \\
&=    \frac{\sqrt{2}}{a\sqrt{\pi}}\left( -\exp(-x^2/2) \right)^\infty_a \\
&=    \frac{\sqrt{2}}{a\sqrt{\pi}}e^{-a^2/2}.
\end{align}
