Positive ensemble mean with a negative realization

Let $\mathrm{\Omega }=\{{\omega }_{+},{\omega }_{-}\}$ with weights $w({\omega }_{+})=3\mathrm{/}4$ and $w({\omega }_{-})=1\mathrm{/}4$. Both weights are nonnegative and sum to $1$, so $(\mathrm{\Omega },w)$ is a finite probability space as defined in Finite probability spaces, outcome weights, events, and event probabilities. For every event $A\subseteq \mathrm{\Omega }$, write $P(A)={\sum }_{\omega \in A}w(\omega )$ for its event probability. Let $Y:\mathrm{\Omega }\to \mathbb{R}$ be given by $Y({\omega }_{+})=2a$ and $Y({\omega }_{-})=-2a$, where $a>0$ and the values are numerical values in the unit $\mathrm{J}{\mathrm{K}}^{-1}$. By Real random variables on finite probability spaces and their finite distributions, $Y$ is a finite-valued real random variable on this finite probability space, so its expectation is a finite sum. The label $Y$ is only an illustrative entropy-valued variable; this example adopts no physical path law.

This arithmetic example illustrates the distinction in Hypotheses for microscopic fluctuation claims.