Let Z (Z,..., Zp) form a random vector where each variable has a N(0, 1) dis- tribution. Let be a positive definite symmetric square matrix and = TT be the Cholesky decomposition of where T is an upper triangular matrix. (You may also find an alternative formulation using the lower triangular matrix, i.e., E = LLT. However, for this problem use the upper triangular formation.) i) (10 points) We want to generate observations from N(, E) using the transfor- mation X = TZ+. Show that, if Z,..., Z,~ N(0, 1), then X~ N(1, 2), using Cholesky decom- position of , where ERP and E RPXP. Find E(X) and Var(X). ii) (10 points) Let 4 -1.5 0.8 and E = -1.5 1 -0.1 0.8 -0.1 1 = (002) Using R, generate n = 1000 observations from N3(4, E) using the Cholesky de- composition technique. Do not use the built-in R function. Make a scatterplot matrix to visualize all the bivariate relationships between the variables. [Hint: You can use the R command chol to perform the Cholesky decomposition. You may notice that it outputs an upper triangular matrix.]