## Find the ERM density

Suppose that $X_1,X_2,...,X_n$ are i.i.d. to $N( \mu , \sigma ^2 )$ with both parameters unknown, we wish to estimate the density function of X with a family of densities as learners and the negative log loss function $L(f) = - \log (f)$

Find the ERM density.

So I know I have to find $\^ {R} (f_{ERM}) = argmin \frac {1}{n} \sum ^n_{k=1} ( - \log f_{X_k} )$

Now I'm stuck here since I don't know what to do, I thought about taking the derivative and set it equals to zero. But then I get $\sum ^n_{k=1} \frac {1}{f_{X_k}} = 0$

But then how would you solve it? Thanks.