Problem 3.4.2 in Rohatgi's An introduction to probability and statistics asks:
Given a random variable whose moment generating function exists, show that
$$P(mX \gt n^2 + \log(M(t)))< e^{-n^2}$$ for $m>0$, and any $s$.
Given that the problem appears in the same chapter as the Chebyshev inequality, I believe that it might be useful, but I am not sure how to start. Any hints on how to get the problem started?
$P(mX>n^2+ln(M(t)))=P(e^{mX}>e^{n^2}M(t))\leq \frac{E[e^{mX}]}{E[e^{tX}]}e^{-n^2}\leq e^{-n^2}$, if $t\geq m$. (using Markov's inequality)