Is Fisher information well-defined here?

Question

I am given a pdf $f_{\mu ,\sigma ^2}=\frac{1}{x\sqrt{2\pi \sigma ^2}}e^{-\frac{1}{2\sigma ^2}\left(lnx-\mu \right)^2} $ and I need to find maximum likelihood estimators of both $\mu$ and $\sigma^2$ with corresponding Fisher information. My approach is to find log-likelihood function $l$ of each and take two derivatives. $\frac{dll}{d\mu} = 0$ gives me $\mu =ln \prod x_i $ and $\frac{d^2ll}{d^2\mu}$ gives the Fisher information. Can anyone tell me if I am moving to the right direction?

Answer 1

Hint: note that the given distribution is a known law: a lognormal distribution

Then the MLE estimations can be obtained immediately and without any calculation, deriving them form the gaussian model.