Problem 1 (yau-contest 2011)

Let $A$ be an $N$-by-$N$ symmetric positive definite matrix. The conjugate gradient method can be described as follows:

$$
r_0=b-Ax_0,\quad p_0=r_0,\quad x_0=0 \\
\text{FOR } n=0,1,\ldots \\
\alpha_n=\frac{||r_n||_2^2}{p_n^T A p_n} \\
x_{n+1}=x_n+\alpha_n p_n \\
r_{n+1}=r_n-\alpha_n A p_n \\
\beta_n=\frac{r_{k+1}^T A p_k}{p_k^T A p_k} \\
p_{n+1}=r_{n+1}+\beta_n p_n \\
\text{END FOR}
$$

Show
(a) $\alpha_n$ minimizes $f(x_n+\alpha p_n)$ for all $\alpha\in\mathbb R$ where
$$f(x)\equiv \tfrac12 x^T A x - b^T x.$$
(b) $p_i^T r_n=0$ for $i<n$ and $p_i^T A p_j=0$ if $i\ne j$.
(c) $\operatorname{Span}\{p_0,p_1,\ldots,p_{n-1}\}=\operatorname{Span}\{r_0,r_1,\ldots,r_{n-1}\}\equiv K_n$.
(d) $r_n$ is orthogonal to $K_n$.

Problem 2 (yau-contest 2011)

We use the following scheme to solve the PDE $u_t+u_x=0$:
$$u_{j}^{n+1}=a u_{j-2}^{n}+b u_{j-1}^{n}+c u_{j}^{n}$$
where $a,b,c$ are constants which may depend on the CFL number $\lambda=\Delta t/\Delta x$. Here $x_j=j\Delta x$, $t^n=n\Delta t$ and $u_j^n$ is the numerical approximation to the exact solution $u(x_j,t^n)$, with periodic boundary conditions.

(i) Find $a,b,c$ so that the scheme is second order accurate.

(ii) Verify that the scheme you derived in Part (i) is exact (i.e. $u_j^n=u(x_j,t^n)$) if $\lambda=1$ or $\lambda=2$. Does this imply that the scheme is stable for $\lambda\le 2$? If not, find $\lambda_0$ such that the scheme is stable for $\lambda\le \lambda_0$. Recall that a scheme is stable if there exist constants $M$ and $C$, which are independent of the mesh sizes $\Delta x$ and $\Delta t$, such that
$$||u^n||\le M e^{CT}||u^0||$$
for all $\Delta x$, $\Delta t$ and $n$ such that $t^n\le T$. You can use either the $L^\infty$ norm or the $L^2$ norm to prove stability.

Problem 3 (yau-contest 2011)

Let $X$ and $Y$ be independent random variables, identically distributed according to the Normal distribution with mean $0$ and variance $1$, $N(0,1)$.
(a) Find the joint probability density function of $(R,\theta)$, where
$$R=(X^2+Y^2)^{1/2}\quad\text{and}\quad \theta=\arctan(Y/X).$$
(b) Are $R$ and $\theta$ independent? Why, or why not?
(c) Find a function $U$ of $R$ which has the uniform distribution on $(0,1)$, $\operatorname{Unif}(0,1)$.
(d) Find a function $V$ of $\theta$ which is distributed as $\operatorname{Unif}(0,1)$.
(e) Show how to transform two independent observations $U$ and $V$ from $\operatorname{Unif}(0,1)$ into two independent observations $X,Y$ from $N(0,1)$.

Problem 4 (yau-contest 2011)

Let $X$ be a random variable such that $E|X|<\infty$. Show that
$$E[|X-a|]=\inf_{x\in\mathbb R} E[|X-x|],$$
if and only if $a$ is a median of $X$.

Problem 5 (yau-contest 2011)

Let $Y_1,\ldots,Y_n$ be iid observations from the distribution $f(x-\theta)$, where $\theta$ is unknown and $f(\,)$ is probability density function symmetric about zero.

Suppose a priori that $\theta$ has the improper prior $\theta\sim$ Lebesgue (flat) on $(-\infty,\infty)$. Write down the posterior distribution of $\theta$. Provides some arguments to show that this flat prior is noninformative.

Show that with the posterior distribution in (a), a $95\%$ probability interval is also a $95\%$ confidence interval.

Problem 6 (yau-contest 2011)

Suppose we have two independent random samples $\{Y_i,\ i=1,\ldots,n\}$ from Poisson with (unknown) mean $\lambda_1$ and $\{Y_i,\ i=n+1,\ldots,2n\}$ from Poisson with (unknown) mean $\lambda_2$. Let $\theta=\lambda_1/(\lambda_1+\lambda_2)$.

(a) Find an unbiased estimator of $\theta$.
(b) Does your estimator have the minimum variance among all unbiased estimators? If yes, prove it. If not, find one that has the minimum variance (and prove it).
(c) Does the unbiased minimum variance estimator you found attain the Fisher information bound? If yes, show it. If no, why not?