Central Limit Theorem for Poisson Confidence Interval

[dreamingotter]

I'm trying to come up with the confidence interval for lambda for samples X1...Xn from Pois(lambda).
When I calculate the fisher information, I(lambda), for one observation, I get 1/lambda.

So by central limit theorem, I get

How do I turn this into this confidence interval? :

I tried to isolate lambda from the CLT statement above, which did not work.

 

[Romsek]

your pics didn't make it

 

[dreamingotter]

I hope this works?

 

[Romsek]

I get that

[MATH]\hat{\lambda} = \bar{X} = \dfrac 1 n \sum \limits_{k=1}^n X_k\\ \text{and by CLT $\bar{X} \approx N\left(\lambda, \sqrt{\dfrac{\lambda}{n}}\right)$}\\ \text{so $\hat{\lambda} \approx N\left(\lambda, \sqrt{\dfrac{\lambda}{n}}\right)$}\\ \text{and a 95% confidence interval for $\lambda$ is $\left[\bar{X}-1.96 \sqrt{\dfrac{\bar{X}}{n}}, ~\bar{X}+1.96 \sqrt{\dfrac{\bar{X}}{n}}\right]$} [/MATH]
So what you've shown in ci.PNG

 

[dreamingotter]

Thanks for your reply, Romsek!
I understand your approach. Now I'm wondering if you could explain how that connects with this approach:
The example below is for another problem, so the MLE is different. I tried to take this approach though, starting at
(λ_hat-λ)/(λ /sqrt(n)) ~ N(0,1). I could not figure out how to isolate lambda though.