1. Let C be the Frank copula with parameter 0 = 5. You may look up the definition here: https://en . wikipedia. org/wiki/Copula_(probability_theory) It can be readily implemented using available software packages which you can use. Let (X, Y ) be a bivariate random vector such that (i) its copula is C and (ii) both the marginal distributions of X and Y are both N(0, 1). (a) Using a direct Monte Carlo simulation, determine the correlation Cov (X, Y) P= VVar(X) Var(Y) between X and Y. Using a preliminary study if necessary, determine a sample size needed such that the 99% confidence interval is shorter than 0.01, and report one such confidence interval. (b) Let F- be the common quantile function of X and Y. For q E (0, 1), let X(q) = P( Y > F-'(q) | X > F-1(q)). Estimate A(q), as accurately as you can, for q E {0.5, 0.6, 0.7, 0.8, 0.9, 0.95}. This is related to the concept of (upper) tail dependence. (c) Let p be the correlation in (a) [you may use the estimate you get], and let (X, Y) be a bivariate normal random vector with zero mean and covariance matrix Find or estimate the corresponding values A ( q ) = P( Y > F2' ( 9 ) | X > F,' ( 9 ) ) , for the values of q in (b). Comment on your results