I am reading the some notes on probability theory and the notes(Chernoff lower bound) left some gap that I should fill in, however I don't even know how to start yet. Here is the assumption the gives by the notes:
1.$M(s)= E[e^{sX}]<\infty$ $ \forall s \in R $
2.The random variable X is continuous, with $PDF$ $f_{X}$.
3.The random variable X does not admit finite upper and lower bounds ( Formally, $0<F_{X}<1$, for all $x\in R$)
The the note claim that:
a) $\lim\limits_{s \rightarrow \infty} \frac{\log M(s)}{s} = \infty$
b) $M(s) $ is differentiable at every $s$
how the assumption related to these claims , any hint for how to start ?