Let (X, Y, Z)T, t2 0, be a joint-Gaussian process with each component a Brownian motion with means E[X] = -2t E[Y ] = t E[Z ] = -4t variances var(X) = t var(Y ) = 2t var(Z) = 9t and covariances cov(X, Y) = -min (t s) cov(X, z) = -2 min (t. s) cov(Y, z) = 4 min (t. s). Define the process G. = -t - 57 +2Y, t20. (a) Find the distribution of G, [3 marks]. (b) Determine if G, is a Brownian motion [3 marks]. (c) Using R, simulate a sample of G, of size n = 105 and assess the normality of the sample using an appropriate plot [3 marks]. (d) Using R, compute P(G, = 40|X, = -1) [3 marks]. Hint. Use R function pmvnorm from mvtnorm package. (e) Find var(G,|X, = -1) = cov(G2, G,|X, = -1) [3 marks]. (1) Find E [(( ) [X, = -1] [3 marks]