Estimators and confidence interval

Question

Can someone explain me how to solve the following exercise? I don't like to post this kind of question, but in this case I have a really bad theory material and I would greatly appreciate a concrete example.

Given the following sample: S = {40, 80, 40, 60, 0, 40, 20, 40, 60, 100}

1. Calculate an estimator of parameter μ of the underlying Poisson variable
2. Calculate an estimator of parameter λ of the underlying negative exponential variable
3. Give a confidence interval with confidence level 0.9 of both parameters

My attempts

1. The underlying Poisson distribution must be coherent with the sample (i.e. the sample average must be equal to the Poisson distribution expected value). $μ=\frac{1}{|S|}\sum_{x\in S}x=\frac{40+80+40+60+0+40+20+40+60+100}{10}=48$
2. Given a random variable $X \sim Exponential(λ)$, we have $μ=E[X]=\frac{1}{λ}$, from which we get $λ=\frac{1}{μ}=\frac{1}{48}$
3. ???

Answer 1

You can approximate the confidence interval for $\lambda$ using a chi-squared distribution. Check this article on wikipedia http://en.wikipedia.org/wiki/Poisson_distribution#Confidence_interval.