Problem 1 (yau-contest 2010)

Let $X_1,\cdots,X_n$ be independent and identically distributed random variables with continuous distribution functions $F(x_1),\cdots,F(x_n)$, respectively. Let $Y_1<\cdots<Y_n$ be the order statistics of $X_1,\cdots,X_n$. Prove that $Z_j=F(Y_j)$ has the beta $(j, n-j+1)$ distribution ($j=1,\cdots,n$).

Problem 2 (yau-contest 2010)

Let $X_1,\cdots,X_n$ be i.i.d. random variable with a continuous density $f$ at point $0$. Let
$$Y_{n,i}=\frac{3}{4b_n}\big(1- X_i^2/b_n^2\big) I(|X_i|\le b_n).$$
Show that
$$\frac{\sum_{i=1}^n (Y_{n,i}-\mathbb E Y_{n,i})}{\big(b_n\sum_{i=1}^n Y_{n,i}\big)^{1/2}} \xrightarrow{\ \mathcal L\ } N(0,3/5),$$
provided $b_n\to 0$ and $nb_n\to\infty$.

Problem 3 (yau-contest 2010)

Let $X_1,\cdots,X_n$ be independently and indentically distributed random variables with $X_i\sim N(\theta,1)$. Suppose that it is known that $|\theta|\le \tau$, where $\tau$ is given. Show
$$\min_{a_1,\cdots,a_{n+1}}\ \sup_{|\theta|\le\tau}\ \mathbb E\Big(\sum_{i=1}^n a_i X_i + a_{n+1} - \theta\Big)^2 
= \frac{\tau^2 n^{-1}}{\tau^2 + n^{-1}}.$$
Hint: Carefully use the sufficiency principle.

Problem 4 (yau-contest 2010)

The rules for "1 and 1" foul shooting in basketball are as follows. The shooter gets to try to make a basket from the foul line. If he succeeds, he gets another try. More precisely, he make 0 baskets by missing the first time, 1 basket by making the first shot and xsmissing the second one, or 2 baskets by making both shots.

Let $n$ be a fixed integer, and suppose a player gets $n$ tries at "1 and 1" shooting. Let $N_0,\ N_1,$ and $N_2$ be the random variables recording the number of times he makes $0,\ 1,$ or $2$ baskets, respectively. Note that $N_0+N_1+N_2=n$. Suppose that shots are independent Bernoulli trials with probability $p$ for making a basket.

(a) Write down the likelihood for $(N_0, N_1, N_2)$.

(b) Show that the maximum likelihood estimator of $p$ is
$$\hat p=\frac{N_1+2N_2}{N_0+2N_1+2N_2}.$$

(c) Is $\hat p$ an unbiased estimator for $p$? Prove or disprove. (Hint: $\mathbb E\hat p$ is a polynomial in $p$, whose order is higher than $1$ for $p\in(0,1)$.)

(d) Find the asymptotic distribution of $\hat p$ as $n$ tends to $\infty$.

Problem 5 (yau-contest 2010)

When considering finite difference schemes approximating partial differential equations (PDEs), for example, the scheme
$$(1)\quad u_j^{n+1}=u_j^n-\lambda(u_j^n-u_{j-1}^n)$$
where $\lambda=\dfrac{\Delta t}{\Delta x}$, approximating the PDE
$$(2)\quad u_t+u_x=0,$$
we are often interested in stability, namely
$$(3)\quad \|u^n\|\le C\|u^0\|,\qquad n\Delta t\le T,$$
for a constant $C=C(T)$ independent of the time step $\Delta t$ and the spatial mesh size $\Delta x$. Here $\|\cdot\|$ is a given norm, for example the $L^2$ norm or the $L^\infty$ norm, of the numerical solution vector $u^n=(u_1^n,u_2^n,\cdots,u_N^n)$. The mesh points are $x_j=j\Delta x$, $t^n=n\Delta t$, and the numerical solution $u_j^n$ approximates the exact solution $u(x_j,t^n)$ of the PDE (2) with a periodic boundary condition.

(i) Prove that the scheme (1) is stable in the sense of (3) for both the $L^2$ norm and the $L^\infty$ norm under the time step restriction $\lambda\le 1$.

(ii) Since the numerical solution $u^n$ is in a finite dimensional space, Student A argues that the stability (3), once proved for a specific norm $\|\cdot\|_a$, would also automatically hold for any other norm $\|\cdot\|_b$. His argument is based on the equivalency of all norms in a finite dimensional space, namely for any two norms $\|\cdot\|_a$ and $\|\cdot\|_b$ on a finite dimensional space $W$, there exists a constant $\delta>0$ such that
$$\delta\|u\|_b\le\|u\|_a\le\frac{1}{\delta}\|u\|_b.$$
Do you agree with his argument? If yes, please give a detailed proof of the following theorem: If a scheme is stable, namely (3) holds for one particular norm (e.g. the $L^2$ norm), then it is also stable for any other norm. If not, please explain the mistake made by Student A.

Problem unknown (yau-contest 2010)

(6) $u_t + C u_x = 0$, $\; C = A + B$.

Here $u$ is a vector of size $m$ and $A$ and $B$ are $m \times m$ real matrices. We assume $m \ge 2$ and both $A$ and $B$ are diagonalizable with only real eigenvalues. We also assume periodic initial condition for these PDEs.

(i) Prove that (4) and (5) are both well-posed in the $L^2$-norm. Recall that a PDE is well-posed if its solution satisfies
$$\|u(\cdot, t)\| \le C(T)\|u(\cdot,0)\|, \quad 0 \le t \le T,$$
for a constant $C(T)$ which depends only on $T$.

(ii) Is (6) guaranteed to be well-posed as well? If yes, give a proof; if not, give a counter example.

(iii) Suppose we have a finite difference scheme $u^{n+1} = A_h u^n$ for approximating (4) and another scheme $u^{n+1} = B_h u^n$ for approximating (5). Suppose both schemes are stable in the $L^2$-norm, namely (3) holds for both schemes. If we now form the splitting scheme $u^{n+1} = B_h A_h u^n$ which is a consistent scheme for solving (6), is this scheme guaranteed to be $L^2$ stable as well? If yes, give a proof; if not, give a counter example.