(a) Show formally that the median of a lognormal distribution with parameters \(\mu\) and \(\sigma\) is \(\mathrm{e}^{\mu}\).

Now generate 1,000 numbers that simulate a lognormal distribution with parameters 3 and 2. (Note that these parameters, while they are often denoted as \(\mu\) and \(\sigma\), are not the mean and standard deviation.)

(b) Compute the mean, variance, and median of your sample. What are the mean, variance, and median of the lognormal distribution with those parameters?

(c) Now, make a \(\log\) transformation on the 1,000 observations to make a new dataset, \(X_{1}, \ldots, X_{1000}\). Make a q-q plot with a normal distribution as the reference distribution.

(d) What are the theoretical mean and variance of the transformed data in Exercise 3.12c?

(e) Now, generate 1,000 numbers from a normal distribution with the same mean and variance as the sample mean and sample variance of the log-transformed data and make a graph of the histograms of the normal sample and the log-transformed data superimposed. Also make a q-q plot of the two samples.