3. Rosenblatt transformation Re maps a vector u E Rd into another vector v E Rd such that Re(u) = v where 01 = U1, 12 = C211 (u2|41), ... Va = Call,.,d_1 (ualu1, ..., Wa-1). (a) If U is a d-dimensional uniform random vector with copula C, then Re(U) is a d-dimensional uniform random vector with independent copula. Prove this result for dimension 2. (b) For the inverse Rosenblatt transformation Re (u) = v where v1 = u1 and Uk = CHIL, k_1 (Ugu1 , ..., UK-1), k = 2, ...,d. If U is a d-dimensional uniform random vector with with independent copula, then Re-'(U) is a d-dimensional uniform random vector with copula C. Prove this result for dimension 2. 4. Let X, Y be any two real-valued random variables. . Show that the regression coefficients a and b that minimise the squared distance E[(Y - (ax + b))?] are given by Cov[X, Y] b* = E[Y] - a*E[X] o2 (X) . Show that the Pearson's coefficient o' (Y) - mina, BE[(Y - (ax + b))2] 1 = ( Y )