CI for Poisson square root Question:
for $x_1...x_n \sim Pois(\lambda)$


*

*Find a CI for $\lambda$ (using the MLE).

*Find a CI for $\psi=\sqrt \lambda$
What I did:
On 1, I found that the MLE is $\bar x$ and then using the normal approximation (the only thing we know how to do in this course) I calculated $\mu=\lambda, \sigma=\sqrt \frac \lambda n$
so the $CI = [\hat \lambda \pm Z_{1-0.5a}\sqrt{\frac {\hat \lambda} n}] $. I want to verify the correctness of the CI. and also - I don't really know how to do part 2. I cannot use linearity of the expectation for calculating it...
 A: Well, you've gone about 1 correctly. It's important to mention that the Poisson RV is a regular exponential family and $\lambda$ is the natural parameterization. The confidence interval for $\sqrt{\lambda}$ can be derived one of two ways, depending on how you calculated the CI in 1. The direct-force method would be reparameterizing the Poisson density in terms of $\psi$, then calculating the MLE and its information. The easy way of going about this is using the Delta method.
A: I believe your interval for part (1) is actually wrong (though almost correct). It's the $Z$ that looks off. Take a careful look; the subscript is wrong in at least one way and probably wrong in a couple of ways.
On to the next part:
If $g$ is a monotonic increasing function, and $(l,u)$ is a $1−α$ interval for $θ$, then $(g(l),g(u))$ will be a $1−α$ interval for $g(θ)$. 
(You can prove it inherits the same coverage by simple transformation of the probability statement for the original CI, but I don't see that you're expected to prove it here)
That is, you can write a confidence interval for $\sqrt \lambda$ immediately, though it won't be symmetric about the transformed parameter estimate (which doesn't strike me as a problem).
The alternative approach would be to write down a normal approximation for the square root of a Poisson, but it's a lot more work. On the other hand, the normal approximation is a lot better for the transformed Poisson.
