LetX1,...,Xnbe a random sample from the lognormal distribution, Lognormal(?,?), whose pdf is:

2. Let X1, ..., Xn be a random sample from the lognormal distribution, Lognormal(M, )), whose pdf is: f (x | M, X ) = 1 xV2TX exp (In ac - 1)2 21 x > 0. (a) Show that the MLE of u and A are u = 1 EL, In X; and 1 = 1 EL, (In Xi - /)2. (b) It is known that In Xi ~ N( u, 1). i. Derive the standard deviation of 1. ii. Derive a 100 . (1 - a)% confidence interval for 1. (c) (R) Consider the following dataset: 12.9 2.3 2.4 65.0 6.7 1.8 1.5 1.7 248.7 1.0 2.0 4.9 3.6 4. 1 6.8 i. Calculate the standard error of 1. ii. Assuming a lognormal distribution is an appropriate model for these data, com- pute the maximum likelihood estimate of A and give a 95% CI. Hint: Quantiles of the lognormal distribution can be computed using the qlnorm () function