2.9.15 Let x = (x1, x2, x3)' be distributed as N3(μ, Σ) where μ = (5, 10, 2)' and
Σ =
$$\begin{pmatrix}
4 & 1 & -1 \\
1 & 2 & 0 \\
-1 & 0 & 1
\end{pmatrix}$$
(a) What is the conditional distribution of x2 and x3 given x1?
(b) What is the conditional distribution of (x1 + x2, x1 - x2) given x3?
(c) What is the distribution of Ax, where A =
$$\begin{pmatrix}
\frac{1}{2} & -1 & \frac{1}{2} \\
-\frac{1}{2} & 0 & \frac{1}{2}
\end{pmatrix}$$ ?
2.9.16 Let x = (x1, x2, x3)' be a vector of the weekly weight of a rat for three weeks.
Suppose x ~ N3(μ, Σ) where μ= (5, 4, 3)' and
Σ =
$$\begin{pmatrix}
2 & 1 & 0 \\
1 & 2 & 1 \\
0 & 1 & 2
\end{pmatrix}$$
What is the distribution of the two weekly weight gains given the initial first
week's weight or equivalently, what is the distribution of (x2 - x1, x3 - x2),
given x1? Are x2 - x1, x3 - x2, and x1 independently distributed?
