Please answer both questions thoroughly and show work! Much appreciated!!

1) Prove the identity: Var(aZ + bZ') = a2Var(Z) + b2Var(Z') + 2abCov(Z, Z'), for any random variables Z, Z' and real numbers a, b, where Var(Z) = E[(Z - E[Z])2] denotes the variance, and Cov(Z, Z')=E[(Z - E[Z])(Z' - E[Z'])] denotes the covariance. 2) We say that a collection of N random variables Z1, ..., ZN (we will assume here that these r.v.'s are either all discrete or all continuous) is mutually independent if we have fZ1..,ZN (Z1, ..., ZN) = fz, (21) ... fZN (ZN), for all admissible choices of the z1, ..., ZN, where the fz,, ... . fzy, and fz1,...,ZN are probability density functions if Z1, ..., Zy are continuous r.v.'s and are probability mass functions if the Z1, ..., ZN are discrete r.v.'s. Moreover, these N variables Z1, ..., ZN are instead said to be pairwise independent if any two variables chosen from this collection of N r.v.'s are independent. a) Does pairwise independence imply mutual independence? Prove or disprove. b) Does mutual independence imply pairwise independence? Prove or disprove