Problem 1 (yau-contest 2012)
Section: Probability and Statistics

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

1) Assume that each $X_n$ satisfies the exponential distribution with parameter $1$ (i.e. $P(X_n \ge x) = e^{-x},\, x \ge 0$). Prove that
(a) $P(X_n > \alpha \log n, \text{i.o.}) = 0$, if $\alpha > 1$; $P(X_n > \alpha \log n, \text{i.o.}) = 1$, if $\alpha \le 1$.
Here “i.o.” stands for “infinitely often”, and $A_n, \text{i.o.}$ stands $\limsup_{n\to\infty} A_n$.
(b) Let $L = \limsup_{n\to\infty} (X_n/\log n)$, then $P(L = 1) = 1$.

2) Assume that each $X_n$ satisfies the Poisson distribution with parameter $\lambda$ (i.e. $P(X_n = k) = \frac{\lambda^k}{k!} e^{-\lambda},\ k = 0,1,2,\dots$). Put
$$L = \limsup_{n\to\infty} \big(X_n \log\log n / \log n\big).$$
Prove that $P(L = 1) = 1$.

Problem 2 (yau-contest 2012)
Section: Probability and Statistics

Let $X_i$ be i.i.d. exponential r.v. with rate one, $i \ge 1$. Let $N$ be a geometric random variable with success probability $p$, $0 < p < 1$, i.e.
$$P(N = k) = (1-p)p^{k-1},\ k = 1,2,\dots,$$
and independent of all $X_i$, $i \ge 1$. Find the distribution of $\sum_{i=1}^{N} X_i$.

Problem 3 (yau-contest 2012)
Section: Probability and Statistics

Let $X$ and $Y$ be i.i.d. real valued r.v.’s. Prove that $$P(|X + Y| < 1) \le 3\, P(|X - Y| < 1).$$

Problem 4 (yau-contest 2012)
Section: Probability and Statistics

Suppose $S = X_1 + X_2 + \cdots + X_n$, a sum of independent random variables with $X_i$ distributed $\operatorname{Binomial}(1, p_i)$. Show that $$\mathbb{P}(S \text{ even}) = \tfrac{1}{2}$$ if and only if at least one $p_i$ equals $\tfrac{1}{2}$.

Problem 5 (yau-contest 2012)
Section: Probability and Statistics

Let $B_\theta$ denote the closed unit ball in $\mathbb{R}^2$ with center $\theta$. Suppose $X_1, X_2, \cdots, X_n$ are independently and uniformly distributed on $B_\theta$, for an unknown $\theta$ in $\mathbb{R}^2$. Denote that maximum likelihood estimator by $\hat{\theta}$. Show that $$|\hat{\theta} - \theta| = O_p(1/n).$$

Problem 6 (yau-contest 2012)
Section: 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) Derive the maximum likelihood ratio test statistic for
$$H_0 : p_1 = p_2 \longleftrightarrow H_1 : p_1 \ne p_2.$$
(Note: No simplification of the resulting test statistic is required. However, you need to give the asymptotic null.)
(b) 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 $\hat{p} = 0.5\,\hat{p}_1 + 0.5\,\hat{p}_2$.