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For any random variable $X$, and any real number $\theta \geq 0$, you have the inequality $$ P(X \geq x) \leq E[e^{\theta X}]/e^{\theta x} = e^{-\theta x} M(\theta) $$

Here $M(\theta)$ is the exponential-generating function for $X$.

Consequently you have Chebyshev's exponential inequality $$ P(X \geq x) \leq \inf_\theta e^{-\theta x} M(\theta) $$

How tight is this formula? While it gives the right asymptotic bounds on $P(X \geq x)$, what if one wants more exact (non-asymptotic) information? Are there any other functions of $M$ that give tighter bounds on $P(X \geq x)$?

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