Problem 1 (yau-contest 2010)

Let $Z_1, \cdots, Z_n$ be i.i.d. random variables with $Z_i \sim N(\mu, \sigma^2)$. Find
$$E\Big(\sum_{i=1}^n Z_i \,\big|\, Z_1 - Z_2 + Z_3\Big).$$

Problem 2 (yau-contest 2010)

Let $X_1, \cdots, X_n$ be pairwise independent. Further, assume that $EX_i = 1 + i^{-1}$ and that $\max_{1 \le i \le n} E|X_i|^{1+\epsilon} < \infty$ for some $\epsilon > 0$. Show that
$$\frac{1}{n} \sum_{i=1}^n X_i \overset{p}{\longrightarrow} 1.$$

Problem 3 (yau-contest 2010)

Let $Z_1, \cdots, Z_6$ be i.i.d. random variables with $Z_i \sim N(0,1)$. Set
$$U^2 = \frac{(Z_1 Z_2 + Z_3 Z_4 + Z_5 Z_6)^2}{Z_2^2 + Z_4^2 + Z_6^2}, \qquad V^2 = \frac{U^2 (Z_2^2 + Z_4^2)}{U^2 + Z_6^2}.$$
Find and identify the densities of $U^2$ and $V^2$.

Problem 4 (yau-contest 2010)

Suppose that three characteristics in a large propulation can be observed according to the following frequencies
$$p_1 = \theta^3, \quad p_2 = 3\theta(1-\theta), \quad p_3 = (1-\theta)^3,$$
where $\theta \in (0,1)$. Let $N_j$, $j=1,2,3$ be the observed frequencies of characteristic $j$ in a random sample of size $n$.
(a) Construct the approximate level $(1-\alpha)$ maximum likelihood confidence set for $\theta$.
(b) Derive the asymptotic distribution for the frequency substitution estimator $\hat{\theta}_2 = 1 - (N_3/n)^{1/3}.$

Problem 5 (yau-contest 2010)

(1) Suppose
$$
S = \begin{bmatrix} \sigma & u^T \\ 0 & S_c \end{bmatrix}, \quad
T = \begin{bmatrix} \tau & v^T \\ 0 & T_c \end{bmatrix}, \quad
b = \begin{bmatrix} \beta \\ b_c \end{bmatrix},
$$
where $\sigma$, $\tau$ and $\beta$ are scalars, $S_c$ and $T_c$ are $n$-by-$n$ matrices, and $b_c$ is an $n$-vector. Show that if there exists a vector $x_c$ such that
$$(S_c T_c - \lambda I)x_c = b_c$$
and $w_c = T x_c$ is available, then
$$
x = \begin{bmatrix} \gamma \\ x_c \end{bmatrix}, \qquad
\gamma = \frac{\beta - \sigma v^T x_c - u^T w_c}{\sigma \tau - \lambda}
$$
solves $(ST - \lambda I)x = b$.

(2) Hence or otherwise, derive an $O(n^2)$ algorithm for solving the linear system $(U_1 U_2 - \lambda I)x = b$ where $U_1$ and $U_2$ are $n$-by-$n$ upper triangular matrices, and $(U_1 U_2 - \lambda I)$ is nonsingular. Please write down your algorithm and prove that it is indeed of $O(n^2)$ complexity.

(3) Hence or otherwise, derive an $O(pn^2)$ algorithm for solving the linear system $(U_1 U_2 \cdots U_p - \lambda I)x = b$ where $\{U_i\}_{i=1}^p$ are all $n$-by-$n$ upper triangular matrices, and $(U_1 U_2 \cdots U_p - \lambda I)$ is nonsingular. Please write down your algorithm and prove that it is indeed of $O(pn^2)$ complexity.

Problem 6 (yau-contest 2010)

(1) Let $A \in \mathbb{R}^{m \times n}$, i.e. $A$ is an $m$-by-$n$ real matrix. Show that there exists an $m$-by-$m$ orthogonal matrix $U$ and an $n$-by-$n$ orthogonal matrix $V$ such that
$$U^T A V = \operatorname{diag}(\sigma_1, \sigma_2, \ldots, \sigma_p),$$
where $p = \min\{m, n\}$ and
$$\sigma_1 \ge \sigma_2 \ge \cdots \ge \sigma_p \ge 0.$$

(2) Let $\operatorname{rank}(A) = r$. Show that for any positive integer $k < r$, 
$$\min_{\operatorname{rank}(B)=k} \|A - B\|_2 = \sigma_{k+1}.$$
(Hint: Consider the matrix $A_k = \sum_{i=1}^k \sigma_i u_i v_i^T$, where $u_i$ and $v_i$ are columns of $U$ and $V$ respectively.)