Let $ \gamma = \frac{1}{\sum_{y}f(y)W(y)}, $
where
$ f(y) = 1 + e^{-|y|} $
and $W(y)$ is a probability distribution (unknown) with $y \in \mathcal{Y}$ arbitrary but discrete, and $x \in \{0,1\}$. I want to calculate a lower bound on $\gamma$. I came up with one lower bound as follows: \begin{aligned} \gamma &= \frac{1}{\sum_{y}f(y)W(y)}\\ &\ge \frac{1}{\sum_{y}f(y)}, \because W(y) \le 1, \forall y \in \mathcal{Y}\\ \end{aligned} I wanted a tighter bond than this. Any ideas or references are appreciated. Thank you !