Question 6 In this question assume that {B(t), t = 0} is the standard Brownian motion (BM). (a) Find P{B(1) + B(2) > 2}. (b) For 0 < s < t, write a formula for the conditional p.d.f. of B(t) given B(s) = x. (c) For 0 < s < t, write a formula for the conditional p.d.f. of B(s) given B(t) = x. d) Using conditions defining the BM,give a detailed proof of the Brownian scaling property: {X(t), t 0} defined by X(t) = cB(t/c) is also a BM, for c > 0 fixed. (e) Which of the following defines a BM: 2B(t/4), tB(1), B(2t) B(t), B(t + 1) B(1)? (f) For the geometric BM {X(t) = e B(t) , t 0}, calculate E[X(t)] and V ar[X(t)].