Exercises 101 in Chapter 4 Of the book "Mathematical Statistics and Data Analysis" by Rica states:
Find the approximate mean and variance of Y = √ X, where X is a random variable following a Poisson distribution.
Here is my attempted solution:
$Y=g(x)=\sqrt{x}$; $ f(x)= e^{-\lambda} \frac{\lambda ^x}{x!}$ where $\mu_x=E[X]=\lambda$ And $\sigma^2_X=Var[X]=\lambda$
I applied the delta method for approximate moments, which states:
$$ \mu_y \approx g(\mu) + \frac{1}{2} \sigma^2_X g``(\mu) = \sqrt{\lambda} + \frac12 \lambda \times(-\frac14 \lambda ^\frac{-3}{2}) = \sqrt{\lambda}-\frac{1}{2\sqrt{\lambda}}$$
$$ \sigma_Y^2 \approx \sigma_X^2[g`(\mu)]^2$$
Is this correct?
Also, in the scripts from my course is that it holds that
$$ Ps(\alpha) \approx N(\alpha,\alpha)$$
I was wondering if this could also be applied in the context of delta method, such as:
$Y=g(x)=\sqrt{x}$; $ f(x) = \frac{1}{\sqrt{2\pi}\lambda}exp{(-\frac{1}{2\lambda^2}\times(x-\lambda)^2})$ where $\mu_x=E[X]=\lambda$ And $\sigma^2_X=Var[X]=\lambda$