3.17. Show that, if X and Z are independent, then each component of X is (px1) (9x1) independent of each component of Z. Hint: P[X, S x1, X2 S x2, .. ., Xp S xp and Z1 S z1, ..., Zq S za] = P[X] S x1, X2 S x2, . . ., Xp S x ] . P[Z, S z1 , ..., Z, s zy] by independence. Let x2, ..., xp and z2, ..., zo tend to infinity, to obtain P[X ] s x, and Z] s z] = P[X, s x]] . P[Z, s z)] for all x1, Z1. So X, and Z, are independent. Repeat for other pairs