matrix algebra and random vectors
Let U1, U2, and U3 be independent random variables, all with the normal distribution with mean zero and standard deviation one. We do not directly observe these random variables, however. Instead, we observe X1, X2, and X3, which are related to U1,U2, and U3 as follows:
X1 = U1 + U2 + ε₁
X2 = U2 + U3 + ε₂
X3 = U3 + U1 + ε₃
where ε₁, ε₂, and ε₃ are independent random variables having the normal distribution with mean zero and standard deviation two.
a) Find the mean vector, μ, and covariance matrix, ∑, for [X1, X2, X3]'
b) Find ∑-1
Hint: From symmetry, the diagonal elements of ∑-1 must all be the same,
and also the off-diagonal elements must all be the same.
c) Find the conditional distribution of X3 given X1 = x1 and X2 = x2.