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2. Let $X = (X, Y)^T$ be multivariate Gaussian with $\mu = EX = \begin{pmatrix} 0 \ 1 \end{pmatrix}$, $\Sigma = Var(X) = \begin{pmatrix} 1 & 1/2 \ 1/2 & 1 \end{pmatrix}$. (i) (4 pts) Determine the conditional distribution of X given Y. (ii) (3 pts) Determine the conditional distribution of 2X + 1 given Y. (iii) (3 pts) Compute $S = \Sigma^{1/2}$. (iv) (3 pts) Find the distribution of $Z = S^{-1}(X - \mu)$. (v) (2 pts) Find the distribution of $||Z||^2$. (vi) (4 pts) Find a linear combination $a_1X + a_2Y = a^TX$ that maximizes $Var(a^TX)/||a||^2$. What is the maximum?

          2. Let $X = (X, Y)^T$ be multivariate Gaussian with
$\mu = EX = \begin{pmatrix} 0 \ 1 \end{pmatrix}$, $\Sigma = Var(X) = \begin{pmatrix} 1 & 1/2 \ 1/2 & 1 \end{pmatrix}$.
(i) (4 pts) Determine the conditional distribution of X given Y.
(ii) (3 pts) Determine the conditional distribution of 2X + 1 given Y.
(iii) (3 pts) Compute $S = \Sigma^{1/2}$.
(iv) (3 pts) Find the distribution of $Z = S^{-1}(X - \mu)$.
(v) (2 pts) Find the distribution of $||Z||^2$.
(vi) (4 pts) Find a linear combination $a_1X + a_2Y = a^TX$ that maximizes $Var(a^TX)/||a||^2$.
What is the maximum?

2. Let X = (X, Y)^T be multivariate Gaussian with
μ = EX =
< p m a t r i x >, Σ = Var(X) =
< p m a t r i x >.
(i) (4 pts) Determine the conditional distribution of X given Y.
(ii) (3 pts) Determine the conditional distribution of 2X + 1 given Y.
(iii) (3 pts) Compute S = Σ^1/2.
(iv) (3 pts) Find the distribution of Z = S^-1(X - μ).
(v) (2 pts) Find the distribution of ||Z||^2.
(vi) (4 pts) Find a linear combination a1X + a2Y = a^TX that maximizes Var(a^TX)/||a||^2.
What is the maximum?

Added by Kaylee A.

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