Problem 1 (yau-contest 2015)
Section: Probability and Statistics

One hundred passengers board a plane with exactly 100 seats. The first passenger takes a seat at random. The second passenger takes his own seat if it is available, otherwise he takes at random a seat among the available ones. The third passenger takes his own seat if it is available, otherwise he takes at random a seat among the available ones. This process continues until all the 100 passengers have boarded the plane. What is the probability that the last passenger takes his own seat?

Problem 2 (yau-contest 2015)
Section: Probability and Statistics

Assume a sequence of random variables $X_n$ converges in distribution to a random variable $X$. Let $\{N_t, t\ge 0\}$ be a set of positive integer-valued random variables, which is independent of $(X_n)$ and converges in probability to $\infty$ as $t\to\infty$. Prove that $X_{N_t}$ converges in distribution to $X$ as $t\to\infty$.

Problem 3 (yau-contest 2015)
Section: Probability and Statistics

Suppose $T_1, T_2,\ldots, T_n$ is a sequence of independent, identically distributed random variables with the exponential distribution of the density function

$$
p(x)=\begin{cases}
 e^{-x}, & x\ge 0; \\
 0, & x<0.
\end{cases}
$$

Let $S_n=T_1+T_2+\cdots+T_n$. Find the distribution of the random vector

$$V_n=\left\{\frac{T_1}{S_n},\ \frac{T_2}{S_n},\ \cdots,\ \frac{T_n}{S_n}\right\}.$$

Problem 4 (yau-contest 2015)
Section: Probability and Statistics

Suppose that $X$ and $Z$ are jointly normal with mean zero and standard deviation $1$. For a strictly monotonic function $f(\cdot)$, $\operatorname{cov}(X,Z)=0$ if and only if $\operatorname{cov}(X,f(Z))=0$, provided the latter covariance exists. Hint: $Z$ can be expressed as $Z=\rho X+\varepsilon$ where $X$ and $\varepsilon$ are independent and $\varepsilon\sim N(0, \sqrt{1-\rho^2})$.

Problem 5 (yau-contest 2015)
Section: Probability and Statistics

Consider the following penalized least-squares problem (Lasso):

$$\frac{1}{2}\|Y-X\beta\|^2+\lambda\|\beta\|_1.$$

Let $\hat{\beta}$ be a minimizer and $\boldsymbol{\Delta}=\hat{\beta}-\beta^*$ for any given $\beta^*$. If $\lambda>2\|X^T(Y-X\beta^*)\|_\infty$, show that

1. $\|Y-X\hat{\beta}\|^2-\|Y-X\beta^*\|^2> -\lambda\|\boldsymbol{\Delta}\|_1$.
2. $\|\boldsymbol{\Delta}_{S^c}\|_1\le 3\|\boldsymbol{\Delta}_S\|_1$, where $S=\{j: \beta_j^*\ne 0\}$ is the support of the vector $\beta^*$, $S^c$ is its complement set, $\boldsymbol{\Delta}_S$ is the subvector of $\boldsymbol{\Delta}$ restricted on the set $S$, and $\|\boldsymbol{\Delta}_S\|_1$ is its $L_1$-norm.