The log-normal distribution is used to model random variables that can only take on positive values and whose logarithms are normally distributed. Thus, a random variable X is said to be log-normally distributed with parameters ,1; and 02 if Y = ln(X) is normally distributed with mean u and variance 0". Log~normal distributions can be used to model returns from a series of trades, the abundanoe of species in ecosystems, and the distribution of minerals in the earth's crust, for example. The p.d.f. of a log-normal distribution is given by 1 I 3 _ (lnzu) 120 a: e H ) m/era" Answer the following questions: (a) List three key properties of maximum likelihood estimators. (3) (b) Show that the maximum likelihood estimates (MLE) for u and of are A = and A2 Zizi (In xi - p)2 n n (7)\f(d) Find approximate 95% confidence intervals for u and o2. (3)(e) Explain how you would use Excel Solver to find maximum likelihood estimates for u and o2 numerically. Be precise about the steps involved