Onstruct a vector of 100 increasing and regularly spaced

Question

The goal of this problem is to prove rigorously the theoretical result illustrated by the simulations of Problem 1.

1. Compute the density of a random variable X whose logarithm log X is N(µ, σ²). Such a random variable is usually called a lognormal random variable with mean µ and variance σ^s.

Throughout the rest of the problem we assume that X is a lognormal random variable with parameters 0 and 1 (i.e. X is the exponential of an N(0, 1) random variable) and that Y is a lognormal random variable with parameters 0 and σ²(i.e. Y is the exponential of an N(0, σ²) random variable). Moreover, we use the notation ρ_min and ρ_max introduced in the last paragraph of Subsection 2.1.2

Problem 1

1. Construct a vector of 100 increasing and regularly spaced numbers starting from .1 and ending at 20. Call it SIG2. Construct a vector of 21 increasing and regularly spaced numbers starting from -1.0 and ending at 1.0. Call it RHO.

2. For each entry σ2 of SIG2 and for each entry ρ of RHO:

• Generate a sample of size N = 500 from the distribution of a bivariate normal vector Z = (X, Y ), where X ∼ N(0, 1), and Y ∼ N(0, σ²), and the correlation coefficient of X and Y is ρ (the S object you create to hold the values of the sample of Z's should be a 500×2 matrix);
• Create a 500 × 2 matrix, call it EXPZ, with the exponentials of the entries of Z (the distributions of these columns are lognormal as defined in Problem 2.7);
• Compute the correlation coefficient, call it ρ˜, of the two columns of EXPZ 3. Produce a scatterplot of all the points (σ², ρ˜) so obtained. Comment.