Problem 1 (yau-contest 2013)

Let $(X_n)$ be a sequence of random variables.

(1) Assume that $\sum_{n=0}^{\infty} P(|X_n| > n) < \infty$. Prove that $\limsup_{n\to\infty} \frac{|X_n|}{n} \le 1$.

(2) Prove that $(X_n)$ converges in probability to $0$ if and only if for certain $r>0$, $E\left[ \frac{|X_n|^r}{1 + |X_n|^r} \right] \to 0$.

Problem 2 (yau-contest 2013)

Let $X$ and $Y$ be independent $N(0,1)$ random variables.

(1) Find $E[X + Y \mid X \ge 0, Y \ge 0]$;

(2) Find the distribution function of $X + Y$ given that $X \ge 0$ and $Y \ge 0$.

(Hint: For b) using the fact that $U = (X + Y)/\sqrt{2}$ and $V = (X - Y)/\sqrt{2}$ are independent and $N(0,1)$ distributed.)

Problem 3 (yau-contest 2013)

Let $\{X_n\}$ be a sequence of independent and identically distributed continuous real valued random variables, and regard $n$ as time. Let $A_n$ be the following event:
$$A_n = \{ X_n = \max\{X_1, X_2, \cdots, X_n\} \}.$$
We say that a maximum record occurs at $n$ in such an event.

(1) Evaluate the probability $P(A_n)$.

(2) Denote by $Y_n$ the number of maximum records occurred until time $n$, i.e.,
$$Y_n = \text{the number of } \{1 \le k \le n : X_k = \max\{X_1, X_2, \cdots, X_k\}\}.$$
Evaluate the expectation $E Y_n$ and the variance $D Y_n$.

Problem 4 (yau-contest 2013)

Let $X = (X_1, \cdots, X_n)$ be an iid sample from an exponential density with mean $\theta$. Consider testing $H_0 : \theta = \theta_0$ vs. $H_1 : \theta > \theta_0$. Let $P(X)$ = your p-value for an appropriate test.

(a) What is $E_{\theta_0}(P(X))$? Derive your answer explicitly.

(b) Derive $E_{\theta}(P(X))$ for $\theta \ne \theta_0$. Specifically, assuming only one sample, i.e. $n = 1$, calculate $E_{\theta}(P(X))$ as explicitly as possible for $\theta \ne \theta_0$.

(c) When there is only one sample, is $E_{\theta}(P(X))$ a decreasing function of $\theta$? In general, can you prove your result for an arbitrary MLR family?

Problem 5 (yau-contest 2013)

Let $X_1, X_2$ be iid uniform on $\theta - \tfrac{1}{2}$ to $\theta + \tfrac{1}{2}$.

(a) Show that for any given $0 < \alpha < 1$, you can find $c > 0$ such that
$$P_{\theta}\{\bar{X} - c < \theta < \bar{X} + c\} = 1 - \alpha,$$
where $\bar{X}$ is the sample mean.

(b) Show that for $\epsilon$ positive and sufficiently small
$$P_{\theta}\{\bar{X} - c < \theta < \bar{X} + c \mid |X_2 - X_1| \ge 1 - \epsilon\} = 1.$$

(c) The statement in (a) is used to assert that $\bar{X} \pm c$ is a $100(1-\alpha)\%$ confidence interval for $\theta$. Does the assertion in (b) contradict this? If your sample observations are $X_1 = 1, X_2 = 2$, would you use the confidence interval in (a)?

Problem 6 (yau-contest 2013)

Suppose you want to estimate the total number of enemy tanks in a war on the basis of the captured tanks. Assume without loss of generality that the tank serial numbers are $1, 2, \cdots, N$, where $N$ is the unknown total number of enemy tanks. Also assume the serial numbers of the $n$ captured tanks are iid uniform on $1, 2, \cdots, N$. (This is a simplified assumption which provides a good approximation if $n \ll N$).

(a) Find the complete sufficient statistic.

(b) Suggest how you may find the minimum variance unbiased estimate of $N$.