1. A random variable follows a lognormal distribution with mean / and variance o' if its log- arithm follows a normal distribution with mean / and variance o', in other words, log X ~ N( H, 02 ). (a) Compute the probability density function of random variable X. Throughout the rest of the problem we assume that X is a lognormal random variable with mean 0 and variance 1 and Y is a lognormal random variable with mean 0 and o'. We further assume that (log X, log Y ) is jointly Gaussian with correlation coefficient p, i.e. p(log X, log Y ) = p. (b) Calculate the correlation between X and Y, p(X, Y). (c) Define Pmin and Pmax as the lower and upper values p(X, Y) can take, i.e., p(X, Y) E [Pmin, Pmax]. Show that Pmin = (e- - 1) / (e -1) (202 - 1) Pmax = (e" - 1) / V(e - 1) (292 - 1)