This is homework!
I have $n$ samples $x_1\dots x_n$ which are i.i.d taken from an Inverse Gaussian distribution $X\sim IG(\mu, \lambda)$.
$f_X(x) = \sqrt{\frac{\lambda}{2\pi x^3}}\exp\bigr({\frac{-\lambda(x-\mu)^2}{2\mu^2x}}\bigl),\space \space \space x \gt 0$
I know that the MLE estimator for $\lambda$ is
$\hat\lambda = \frac{n}{\sum_i^n(\frac{1}{X_i}-\frac{1}{\bar X})}$.
Using the fact that $\frac{n\lambda}{\hat\lambda}\sim\chi_{n-1}^2$, I need to find the $100(1-\alpha)\text{%}$ confidence interval for $\lambda$.
What I did:
I know that $\space\space L, U = \hat\lambda \space\pm\space c\frac{SD(\hat\lambda)}{\sqrt{n}}$
and since that the distribution is not normal and n is unknown, I should probably use $t_{29}(c) = 1 - \frac{\alpha}{2}$, to find c.
I don't know the sample size, so the final answer should include an $n$.
I don't know how to find the standard deviation of $\hat\lambda$.
If also possible, can you show me how would you find the same CI in R.
Thanks in advance.