Problem 1 (yau-contest 2011)
Section: Applied Math., Computational Math., Probability and Statistics

Given a weight function $\rho(x) > 0$, let the inner-product corresponding to $\rho(x)$ be defined as follows:
$(f,g) := \int_a^b \rho(x) f(x) g(x)\,dx,$
and let $\|f\| := (f,f)$.

(1) Define a sequence of polynomials as follows:
$p_0(x) = 1,\quad p_1(x) = x - a_1,$
$p_n(x) = (x - a_n) p_{n-1}(x) - b_n p_{n-2}(x),\quad n = 2, 3, \cdots,$
where
$a_n = \dfrac{(x p_{n-1}, p_{n-1})}{(p_{n-1}, p_{n-1})},\quad n = 1, 2, \cdots,$
$b_n = \dfrac{(x p_{n-1}, p_{n-2})}{(p_{n-2}, p_{n-2})},\quad n = 2, 3, \cdots.$
Show that $\{p_n(x)\}$ is an orthogonal sequence of monic polynomials.

(2) Let $\{q_n(x)\}$ be an orthogonal sequence of monic polynomials corresponding to the $\rho$ inner product. (A polynomial is called monic if its leading coefficient is 1.) Show that $\{q_n(x)\}$ is unique and it minimizes $\|q_n\|$ amongst all monic polynomials of degree $n$.

(3) Hence or otherwise, show that if $\rho(x) = 1/\sqrt{1 - x^2}$ and $[a,b] = [-1,1]$, then the corresponding orthogonal sequence is the Chebyshev polynomials:
$T_n(x) = \cos(n\arccos x),\quad n = 0,1,2,\cdots,$
and the following recurrent formula holds:
$T_{n+1}(x) = 2x T_n(x) - T_{n-1}(x),\quad n = 1,2,\cdots.$

(4) Find the best quadratic approximation to $f(x) = x^3$ on $[-1,1]$ using $\rho(x) = 1/\sqrt{1 - x^2}$.

Problem 2 (yau-contest 2011)
Section: Applied Math., Computational Math., Probability and Statistics

If two polynomials $p(x)$ and $q(x)$, both of fifth degree, satisfy
$p(i) = q(i) = \frac{1}{i},\quad i = 2, 3, 4, 5, 6,$
and
$p(1) = 1,\quad q(1) = 2,$
find $p(0) - q(0)$.

Problem 3 (yau-contest 2011)
Section: Applied Math., Computational Math., Probability and Statistics

Lay aside $m$ black balls and $n$ red balls in a jug. Supposes $1 \le r \le k \le n$. Each time one draws a ball from the jug at random.
1) If each time one draws a ball without return, what is the probability that in the $k$-th time of drawing one obtains exactly the $r$-th red ball?
2) If each time one draws a ball with return, what is the probability that in the first $k$ times of drawings one obtained totally an odd number of red balls?

Problem 4 (yau-contest 2011)
Section: Applied Math., Computational Math., Probability and Statistics

Let $X$ and $Y$ be independent and identically distributed random variables. Show that
$E[|X + Y|] \ge E[|X|].$
Hint: Consider separately two cases: $E[X^+] \ge E[X^-]$ and $E[X^+] < E[X^-].$

Problem 5 (yau-contest 2011)
Section: Applied Math., Computational Math., Probability and Statistics

Suppose that $X_1, \cdots, X_n$ are a random sample from the Bernoulli distribution with probability of success $p_1$ and $Y_1, \cdots, Y_n$ be an independent random sample from the Bernoulli distribution with probability of success $p_2$.
(a) Give a minimum sufficient statistic and the UMVU (uniformly minimum variance unbiased) estimator for $\theta = p_1 - p_2$.
(b) Give the Cramer-Rao bound for the variance of the unbiased estimators for $v(p_1) = p_1(1 - p_1)$ or the UMVU estimator for $v(p_1)$.
(c) Compute the asymptotic power of the test with critical region
$\left|\sqrt{n}(\hat{p}_1 - \hat{p}_2)/\sqrt{2\hat{p}\hat{q}}\right| \ge z_{1-\alpha}$
when $p_1 = p$ and $p_2 = p + n^{-1/2}\Delta$, where $\hat{p} = 0.5\hat{p}_1 + 0.5\hat{p}_2$.

Problem 6 (yau-contest 2011)
Section: Applied Math., Computational Math., Probability and Statistics

Suppose that an experiment is conducted to measure a constant $\theta$. Independent unbiased measurements $y$ of $\theta$ can be made with either of two instruments, both of which measure with normal errors: for $i = 1,2$, instrument $i$ produces independent errors with a $N(0, \sigma_i^2)$ distribution. The two error variances $\sigma_1^2$ and $\sigma_2^2$ are known. When a measurement $y$ is made, a record is kept of the instrument used so that after $n$ measurements the data is $(a_1, y_1), \ldots, (a_n, y_n)$, where $a_m = i$ if $y_m$ is obtained using instrument $i$. The choice between instruments is made independently for each observation in such a way that
$P(a_m = 1) = P(a_m = 2) = 0.5, \quad 1 \le m \le n.$

Problem unknown (yau-contest 2011)

Let $x$ denote the entire set of data available to the statistician, in this case $(a_1, y_1), \ldots, (a_n, y_n)$, and let $l_\theta(x)$ denote the corresponding log likelihood function for $\theta$. Let $a = \sum_{m=1}^n (2 - a_m)$.

(a) Show that the maximum likelihood estimate of $\theta$ is given by
$$\hat{\theta} = \left( \sum_{m=1}^n 1/\sigma_{a_m}^2 \right)^{-1} \left( \sum_{m=1}^n y_m/\sigma_{a_m}^2 \right).$$

(b) Express the expected Fisher information $I_\theta$ and the observed Fisher information $I_x$ in terms of $n$, $\sigma_1^2$, $\sigma_2^2$, and $a$. What happens to the quantity $I_\theta/I_x$ as $n \to \infty$?

(c) Show that $a$ is an ancillary statistic, and that the conditional variance of $\hat{\theta}$ given $a$ equals $1/I_x$. Of the two approximations
$$\hat{\theta} \sim N(\theta, 1/I_\theta)$$
and
$$\hat{\theta} \sim N(\theta, 1/I_x),$$
which (if either) would you use for the purposes of inference, and why?