Approximate values of $\operatorname{E}[\sqrt X]$ and $\operatorname{Var}[\sqrt X]$ for $X$ Poisson distributed with parameter $\lambda\to\infty$

Question

Assume that $X$ has a Poisson distribution with rate parameter $\lambda$. If $Y = \sqrt X$, using moment-generating functions or otherwise, show that $$\operatorname{E}[Y] \approx \sqrt\lambda - \frac 1 {8 \sqrt\lambda}$$ and $$\operatorname{Var}[Y] \approx \frac14$$

A suggestion is to use MGFs but I've got no idea how to go from there as I keep getting jammed.

Answer 1

This is an exercise in the delta method. Since $dY/dX=1/(2\sqrt{X})$, $\operatorname{Var}Y\approx\frac{\lambda}{(2\sqrt{\lambda})^2}=\frac{1}{4}$. Then $E(Y)=\sqrt{\lambda}\sqrt{1-\frac{1}{4\lambda}}$ gives the desired result.