Please help me soslve this question with steps, note I have received some answers from other tutors which are incorrect. Please help me with your own work.

Note that: BB=standard Brownian bridge,BM=standard Brownian motion, GP=Gaussian process

H2 A BB is Bo = (B?) > is a sample-continuous GP, having zero mean mB(t) = 0 and covariance function CBo(s, t) = $ At - st, s,t 2 0. If Z ~ Normal(0, 1) and BM B = (B+)to, verify (a) P(B; = Bi = 0) = 1; (b) WP := Be - tB1, t 2 0, defines BB Wo = (WP)t; (c) if Z and Bo are independent, then Wt := B + Zt, t 2 0, defines BM W = (W.); (d) law(BIB, = 0) =law(Bo) [Hint: determine Gaussian conditional densities]