Problem 1 Let X X1 X2 X3 be N3Mu Sigma with Mu 1 1 2 and S

Problem 1: Let X = (X1, X2, X3)? be N3(Mu, Sigma) with Mu = (1, -1, 2)? and Sigma = [1 -1 0 -1 2 0 0 0 4] a. Find the distribution of X1 + X2. b. Find the conditional distribution of X1, given X2 = x2 c. Find the conditional distribution of X1, given X2 = x2 and X3 = x3. Compare this with [b], comment on your result. d. Find the conditional distribution of X1 + X2, given X1 + X3 = c.

Solution

a) X1+X2 will be a normal distribution with mean =0 and

std dev = 1

b) conditional distribution of x1 = normal distribution with x1 and std dev = root 2

c) Normal distribution with (0,1)

d) If x1+x3 =c,

then c follows a distribution with p(c) =1