Chernoff Inequality

Because, Markov’s inequality may take any monotonically increasing function, the function \(g(X) = e^{tX}\) is a good choice since the RHS has the moment generating function \(E(e^{tX}) = M_X(t)\),

\[ P(X \ge c) \le \frac{M_X(t)}{e^{tc}} \]

Right Tail Bound

Since the MGF is a function of \(t\) we can choose any \(t\) and preferably the one that minimizes the RHS,

\[ \boxed{P(X \ge c) \le \min_{t \gt 0} \frac{M_X(t)}{e^{tc}}} \]

Left Tail Bound

\[ \boxed{P(X \le c) \le \min_{t \gt 0} \frac{M_X(-t)}{e^{-tc}}} \]