Problem 1 (yau-contest 2018)
Section: Probability and Statistics

A submarine is lost in some ocean. There are two (and only two) possible regions: A and B. Experts estimate the probability of being lost in A is 70%. On the other hand, for each search, the probability of finding this submarine is 40% if it is lost in A. This number is 80% for region B. Now we have independently searched region A 4 times and region B once, but still have not found the submarine yet. Now based on these informations, which region we should search next? And why?

Problem 2 (yau-contest 2018)
Section: Probability and Statistics

A teacher and 12 students sit around a circle. In the beginning the teach holds a gift, he will randomly pass it to the left person or right person next to him, so as the other students each time. (For the gift, It is like a random walk between these people) The rule is that the gift will be eventually given to some student (not teacher) if he/she is the last student who ever touches the gift. Which student(s) have the highest probability to get this gift (i.e., win)?

Problem 3 (yau-contest 2018)
Section: Probability and Statistics

In a party, $N$ people attend, each of them brings $k$ gifts. When they leave, each of them randomly picks $k$ gifts. Let $X$ be the total number of gifts which are taken back by their owners. Let’s fix $k$, please find the limiting distribution of $X$ when $N \to \infty$.

Problem 4 (yau-contest 2018)
Section: Probability and Statistics

Suppose that a random vector $\mathbf{x} = (x_1, ..., x_n)' \in R^n\,(n \ge 2)$ is distributed as a multivariate normal distribution $N(0, \Sigma)$ with the following joint probability density function
$$f(\mathbf{x}) = \frac{1}{(2\pi)^{n/2} \det(\Sigma)^{1/2}} \exp\left\{ -\frac{1}{2} \mathbf{x}' \Sigma^{-1} \mathbf{x} \right\}, \quad \mathbf{x} \in R^n,$$
where $\Sigma$ is an $n \times n$ positive definite matrix. Let the $(i, j)$ element of $\Omega = \Sigma^{-1}$ be $\omega_{ij}$ ($1 \le i, j \le n$). For $1 \le i \ne j \le n$, show that if $\omega_{ij} = 0$, then $x_i$ and $x_j$ are conditionally independent when the other elements of $\mathbf{x}$ are given.

Problem 5 (yau-contest 2018)
Section: Probability and Statistics

Let $\mathbf{x}, \mathbf{y}$ be two independent random vectors in $R^n\,(n \ge 3)$. Assume that $P(\mathbf{y} = 0) = 0$ and $\mathbf{x}$ has a standard multivariate normal distribution, i.e., $\mathbf{x} \sim N(0, I_n)$.

(a) For any nonzero constant vector $\mathbf{a} \in R^n$ satisfying $\|\mathbf{a}\| = (\mathbf{a}'\mathbf{a})^{1/2} = 1$, prove that
$$\sqrt{n-1}\,\frac{\mathbf{a}'\mathbf{x}}{\sqrt{\|\mathbf{x}\|^2 - (\mathbf{a}'\mathbf{x})^2}} \sim t_{n-1},$$
here $t_{n-1}$ stands for a $t$ distribution with $n-1$ degrees of freedom.

(b) The sample correlation coefficient between $\mathbf{x} = (x_1, ..., x_n)'$ and $\mathbf{y} = (y_1, ..., y_n)'$ is defined as
$$r = \frac{\sum_{i=1}^n (x_i - \bar{x})(y_i - \bar{y})}{\sqrt{\sum_{i=1}^n (x_i - \bar{x})^2\, \sum_{i=1}^n (y_i - \bar{y})^2}},$$
where $\bar{x} = \sum_{i=1}^n x_i / n$, $\bar{y} = \sum_{i=1}^n y_i / n$. Show that
$$\sqrt{n-2}\,\frac{r}{\sqrt{1-r^2}} \sim t_{n-2}.$$