 Solution

Let $\mu =⟨X⟩$. The Markov jump-process generator applied to $X$ gives the exact moment equation $\dot{\mu }=3\lambda -\beta ⟨\sqrt{X}⟩$. With the prescribed moment closure, $⟨\sqrt{X}⟩\approx \sqrt{\mu }$, stationarity givesThe production flux counts molecules, not reaction events: it is $v^{+}=3\lambda$, while the removal flux is $v^{-}=\beta \sqrt{\mu }=3\lambda$. The reaction-rate elasticity of removal relative to constant production isThe mean molecular lifetime is population divided by its stationary molecular removal flux. ThereforeThe noise-relevant average chemical event size is the molecular-flux-weighted reaction event size. For the stoichiometric vectors $+3$ and $-1$ it isThis weights each absolute jump by the molecular turnover it contributes. Equivalently, the balanced production and removal fluxes give equal weight to their sizes $3$ and $1$, yielding $(3+1)/2$. An event-count-weighted average would be $3/2$, since three quarters of firing events are deaths at stationarity; that different average does not determine the diffusion term in the linear noise approximation.