Let

x

be

N_(3)(\\\\mu ,\\\\Sigma )

with

\\\\mu ^(')=[2,-3,1]

and\

\\\\Sigma =[[1,1,1],[1,3,2],[1,2,2]]

\ (a) Find the distribution of

3x_(1)-2x_(2)+x_(3)

.\ (b) Relabel the variables if necessary, and find a

2\\\\times 1

vector a such that

x_(2)

and

x_(2)-a^(')[[x_(1)],[x_(3)]]

are independent.\ (c) Find the conditional distribution of

x_(3)

, given that

x_(1)=x_(1)

and

x_(2)=x_(2)

.

2. Let X be N3(,) with =[2,3,1] and =111132122 (a) Find the distribution of 3X12X2+X3. (b) Relabel the variables if necessary, and find a 21 vector a such that X2 and X2a[X1X3] are independent. (c) Find the conditional distribution of X3, given that X1=x1 and X2=x2