Just a note: this is a homework question, so feel free to prod me towards the answer if you want :) Also, I'm pretty bad at statistics so sorry in advance if I'm stupid :/
For some $ X_1 ... X_n \sim Exp(\lambda) $ i.i.d. random variables, I have an estimator:
$ \hat{\lambda} = (\frac{1}{n} \sum_{i = 1}^{n}{X_i})^{-1} $
I need to use the CLT to determine an approximate $ 1 - \alpha $ stochastic confidence interval for $ \frac{1}{\lambda} $ based on $ \hat{\lambda} $.
Obviously, I need to find the mean $ \mu $ and variance $ \sigma^2 $ so that I can normalise it (i.e. $ \frac{\hat{\lambda} - \mu}{\sigma} \sim Normal(0, 1) $).
The thing I'm not sure on is how to find $ \mu $ and $ \sigma^2 $.
I'm mostly comfortable saying that since $ \frac{1}{n} \sum_{i = 1}^{n}{X_i} $ is the sample mean, $ E[\hat{\lambda}] = \lambda $
With the variance though, I'm really not sure. My best guess is that it's:
$$ Var(\hat{\lambda}) = (Var(\frac{1}{n} \sum_{i = 1}^{n}{X_i}))^{-1} $$
$$ Var(\hat{\lambda}) = (\frac{1}{n^2} \dot{} Var(\sum_{i = 1}^{n}{X_i}))^{-1} $$
$$ Var(\hat{\lambda}) = (\frac{1}{n^2} \dot{} n \dot{} \frac{1}{\lambda^2} )^{-1} $$
$$ Var(\hat{\lambda}) = n\lambda^2 $$
But I'm not sure if you're do that with variances. Any help would be appreciated :)