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2. (15 points) Let $X_1, \dots, X_n$ be jointly normally distributed with mean $\vec{\mu} = \vec{0}$ and covariance matrix $\Sigma = A^{-1}$ ($A$ is called the precision matrix). That is, $X_1, \dots, X_n$ have joint density $f(x_1, \dots, x_n) = \frac{1}{(2\pi |\Sigma|)^{n/2}} \exp \left( -\frac{1}{2} \sum_{i,j=1}^n x_i A_{ij} x_j \right)$ Show the conditional distribution $f(x_1 | x_2, \dots, x_n)$ is normal and find its mean and variance. Hint: do not do unnecessary work, including unnecessary integrals. The precision matrix is necessarily symmetric: $A_{ij} = A_{ji}$

          2. (15 points) Let $X_1, \dots, X_n$ be jointly normally distributed with mean $\vec{\mu} = \vec{0}$ and covariance matrix $\Sigma = A^{-1}$ ($A$ is called the precision matrix). That is, $X_1, \dots, X_n$ have joint density

$f(x_1, \dots, x_n) = \frac{1}{(2\pi |\Sigma|)^{n/2}} \exp \left( -\frac{1}{2} \sum_{i,j=1}^n x_i A_{ij} x_j \right)$

Show the conditional distribution $f(x_1 | x_2, \dots, x_n)$ is normal and find its mean and variance.
Hint: do not do unnecessary work, including unnecessary integrals. The precision matrix is necessarily symmetric: $A_{ij} = A_{ji}$

2. (15 points) Let X1, …, Xn be jointly normally distributed with mean μ⃗ = 0⃗ and covariance matrix Σ = A^-1 (A is called the precision matrix). That is, X1, …, Xn have joint density

f(x1, …, xn) = (1)/((2π |Σ|)^n/2)( -(1)/(2)∑i,j=1^n xi Aij xj )

Show the conditional distribution f(x1 | x2, …, xn) is normal and find its mean and variance.
Hint: do not do unnecessary work, including unnecessary integrals. The precision matrix is necessarily symmetric: Aij = Aji

Added by Jennifer L.

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