 Solution

For $Y_{\mu }\sim \mathrm{Poisson}(\mu )$, the delta method with $g(y)=2\sqrt{y}$ givesThus the transformation stabilizes variance at large $\mu$.

A Gaussian linear model for $\sqrt{Y}$ treats the transformed observations as having additive constant-variance errors and models $\mathbb{E}(\sqrt{Y}\mid X)$ as linear. A Poisson regression with square-root link retains the Poisson likelihood and modelsThe two procedures therefore have different likelihoods, fitted means, and constraints; transforming the response is not the same as transforming its conditional mean.