C$.$ Let $x_1,$dots,$x_n$ be a sample of a population $x$ with density function

$f(x)=\{\begin{array}{cc}\frac{1}{x\sqrt[\phantom{2}]{2}}{e}^{\frac{-1}{2}(log(x)-{)}^2},& x0\\ 0,& \text{e}\text{l}\text{s}\text{e}\text{w}\text{h}\text{e}\text{r}\text{e}& \end{array}$

with $inR,>0.$

I a$.$ Based on this sample, find the Maximum Likelihood Estimators hat$()$ and hat$()$ for the parameter $$ and $($You do not need to show that they are maximums by using the second order derivatives$).$

b$.$ Find the density of the random variable $Z=log(x).$

c$.$ Give the asymptotic distribution of the Maximum Likelihood Estimator hat$()?$

Give sufficient details in your derivations.