# Lower bound on a function of probability distribution

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 !

-

As $$f(y) = 1 + e^{-|y|}\leq 2$$ then $$E[f(y)]=\sum_{y} f(y)W(y)\leq \sum_{y} 2 ~W(y)=2$$ Hence $$\gamma = \frac{1}{\sum_{y}f(y)W(y)}\geq \frac{1}{2}$$

-
$1 \lt f(y) \le 2$
so $1=\sum_{y}W(y) \lt \sum_{y}f(y)W(y) \le \sum_{y}2W(y) =2$
so $\frac12 \le \gamma \lt 1$.