Let Y=(Y1,Y2,Y3,) be a random vector with mean vector and covariance matrix given by

m = (1 , -1 , 3)

and Cov (Y) = 1 1 0

1 2 3

0 3 10

a. Let Z1= Y1 + Y2 + Y3, Z2 = 3Y1 +Y2 +Y3

Find for Z=(Z1,Z2)

i. E(Z)

ii. Var Z

iii. Cov (Z1,Z2)

Given the information in part (b), determine

i. The correlation matrix for Z

ii. Corr (Z1,Z2)

1. Show that for a matrix A and a variance- covariance matrix (Summation sign)

(Summation sign)Ax= A(Summation sign)At

Suppose that x1 and x2 are independent with density function fx(x1) and fx(x2) find the distribution of u1= x1 + x2 and u2= x1-x2