Problem 1 (yau-contest 2016)
Section: Probability and Statistics

A random walker moves on the lattice $\mathbb{Z}^2$ according to the following rule: in the first step it moves to one of its neighbors with probability $1/4$, and then in step $n>1$ it moves to one of the neighbors that it didn’t visit in the step $n-1$ with equal probability. Let $T$ be the time when the random walker steps on a site that it already visited. Please show that the expectation of $T$ is less than $35$.

Problem 2 (yau-contest 2016)
Section: Probability and Statistics

Let $X$ be a $N \times N$ random matrix with i.i.d. random entries, and $$\mathbb{P}(X_{11} = 1) = \mathbb{P}(X_{11} = -1) = 1/2.$$ Define $$\|X\|_{\mathrm{op}} = \sup_{v \in \mathbb{C}^N:\|v\|_2 = 1} \|Xv\|_2.$$ Please show that for any fixed $\delta > 0$, $$\lim_{N\to\infty} \mathbb{P}\big(\|X\|_{\mathrm{op}} \ge N^{1/2+\delta}\big) = 0.$$ Hint: $\|X\|_{\mathrm{op}}^2 \le \mathrm{tr}|X|^2$.

Problem 3 (yau-contest 2016)
Section: Probability and Statistics

Suppose that $2016$ balls are put into $2016$ boxes with each ball independently being put into box $i$ with probability $\frac{1}{3\times 1008}$ for $i \le 1008$ and $\frac{2}{3\times 1008}$ for $i > 1008$. Let $T$ be the number of boxes containing exactly $2$ balls. Please find the variance of $T$.

Problem 4 (yau-contest 2016)
Section: Probability and Statistics

Let $b>a>0$ be real numbers. Let $X$ be a random variable taking values in $[a,b]$, and let $Y=\frac{1}{X}$. Determine the set of all possible values of $\mathbb{E}(X)\times \mathbb{E}(Y)$.

Problem 5 (yau-contest 2016)
Section: Probability and Statistics

Let $X_1, X_2, \ldots$ be independent and identically distributed real-valued random variables such that $\mathbb{E}(X_1) = -1$. Let $S_n = X_1 + \cdots + X_n$ for all $n \ge 1$, and let $T$ be the total number of $n \ge 1$ satisfying $S_n \ge 0$. Compute $\mathbb{P}(T = \infty)$.