Let X ~ F where F(x) = 1 - c(x +x)", r > u, for some c > 0, KER, 0 > 0 and u > 0. a) Show that, for all on and oz satisfying F(u) 1. Show that, for all on and oz satisfying F(u) 0? c) What is the mean excess function of the exponential distribution?Consider a bivariate distribution function F with density f($1,12) = 0(0 + 1)(21 + 12 -1) (+2), $1, 12 ( [1,00), 0 > 0. a) Derive F. b) Derive the marginal distribution functions Fi, F, of F. c) Compute the correlation coefficient p corresponding to F. What assumption must we make about & in order that p exists?Let E be the positive-definite matrix E= Po102 po102 1919 for 01 > 0, 02 > 0 and pe (-1, 1). Show that E has a unique Cholesky factor A. In other words, show that there exists a unique lower-triangular matrix A with positive diagonal entries satisfying ) = AA