# asymptotic confidence interval of poisson distribution

Find the variance stabilizing transformation $$g$$ that satisfies $$\sqrt{n}(g(\bar{X_{n}})-g(\lambda)) \rightarrow Z, Z \sim N(0,1)$$ where $$\bar{X_{n}}$$ is a sample mean of random variables $$X_{1}, \cdots, X_{n}$$ that follow Poisson distribution $$\operatorname {Poisson}(\lambda)$$. Use the result to find the asymptotic confidence interval in terms of $$\lambda$$.

First, $$\sqrt{n}(\bar{X_{n}}-\lambda) \rightarrow N(0,\lambda)$$ holds from CLT. Then, $$\sqrt{n}(g(\bar{X_{n}})-g(\lambda)) \rightarrow N(0,\{g'(\lambda)\}^{2}\lambda)$$ , which should be equivalent to $$N(0,1)$$, so $$g'(\lambda) = \frac{1}{\sqrt{\lambda}}$$ and $$g(\lambda) = 2\sqrt{\lambda}$$. Therefore, $$\sqrt{n}(\sqrt{\bar{X_{n}}}-\sqrt{\lambda}) \rightarrow N(0,\frac{1}{4}).$$ Now how should I proceed further?

You are almost there!

You correctly found that the "variance stabilizing transformation $$g()$$" is $$2\sqrt{()}$$

that is

$$\sqrt{n}\left[2\sqrt{\overline{X}_n} -2\sqrt{\lambda} \right]\dot{\sim}N(0;1)$$

Now all you have to do is to fix the two Gaussian quantiles with a certain significance level $$\alpha$$, set

$$-z<\sqrt{n}\left[2\sqrt{\overline{X}_n} -2\sqrt{\lambda} \right]

and solve the double inequality in $$\lambda$$