Show the calculations.

Exercise 10 (#1.31). Let F and G be two cumulative distribution func- tions on the real line. Show that if F and G have no common points of discontinuity in the interval [a, b], then G(x)dF(x) = F(b)G(b) - F(a)G(a) - / F(x)dG(x). J(a, b] (a, b]Exercise 8 (#1.22). Let v be a measure on a o-field F on ? and f and q be Borel functions with respect to F. Show that (i) if f fdy exists and a E R, then [(af )du exists and is equal to a f fdy; (ii) if both ffdu and f gdu exist and f fdu + f gdv is well defined, then [(f + g)du exists and is equal to f fdu + f gdu. Note. For integrals in calculus, properties such as [(af)du =a f fdy and [(f + g)dy f fdu + / gdu are obvious. However, the proof of them are complicated for integrals defined on general measure spaces. As shown in this exercise, the proof often has to be broken into several steps: simple functions, nonnegative functions, and then general functions.Exercise 21 (#1.38). Let (X, Y, Z) be a random 3-vector with the fol- lowing Lebesgue density: -sin r sin y sin z f (x, y, 2) = 873 0 otherwise Show that X. Y. Z are pairwise independent, but not independent.Exercise 5. Let X = (X1,.... X,) be a random n-vector having the mul- tivariate normal distribution N, (/J. D), where J is the n-vector of 1's. D = 02 P P p and (p) d N2(0, 2) and vn(71-d) -> 2 N(0, c'). Identify e, E, d, and c' in terms of moments of X1. (ii) Repeat (i) when X] has the normal distribution N(0, o?). (iii) Repeat (i) when X1 has Lebesgue density (20)-1e-lil/6