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An upper bound for the moment generating function $f(t) = \mathbb{E}[e^{tX}]$ when the mean, variance and upper bound of $X$ are given
We first observe the following:
Observation. For each $s > 0$, define the function $f_s(y)$ by
$$ f_s(y) = \frac{1 - e^{-sy}}{y} = \int_{0}^{s} e^{-yu} \, \mathrm{d}u. $$
Then $f_s(y)$ is a ...
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