Problem 1 (qiuzhen-qualify 2025-fall)
Section: Probability & Statistics

(10 points) Let $X$ and $N$ be two independent random variables. Suppose $X$ follows the standard exponential distribution (i.e., with probability density function $e^{-x}\mathbf{1}_{\{x>0\}}$), and $N$ follows the Poisson distribution with parameter $\lambda>0$.

(a) Find all $\lambda>0$ such that $\mathbb{E}[X^N]<\infty$.

(b) Calculate the expectation of $\mathbb{E}[X^N]$ under the above condition on $\lambda$.

(c) Is there any $\lambda>0$ such that $\operatorname{Var}[X^N]<\infty$? Justify your answer.

Problem 2 (qiuzhen-qualify 2025-fall)
Section: Probability & Statistics

(10 points) Let $X,Y$ be two independent random variables uniformly distributed on $[0,1]$, and define
$$Z := \max\{X,Y\}.$$
(a) Determine the distribution of $Z$ conditioned on $\{X+Y \in [0,1]\}$.

(b) Determine the distribution of $Z$ conditioned on $\{X+Y \in [1,2]\}$.

Problem 3 (qiuzhen-qualify 2025-fall)
Section: Probability & Statistics

(10 points) Let $(X_n,\mathcal{F}_n)$ be a martingale.

(a) Prove that if $(X_n^2,\mathcal{F}_n)$ is also a martingale, then $X_m=X_n$ a.s. for any $m,n\in\mathbb{N}$.

(b) Prove that if $(|X_n|^p,\mathcal{F}_n)$ is a martingale for some $p>1$, then $(|X_n|^q,\mathcal{F}_n)$ is also a martingale for every $1\le q\le p$.

Problem 4 (qiuzhen-qualify 2025-fall)
Section: Probability & Statistics

(15 points) Given independent random variables $(X_n)_{n\in\mathbb{N}}$ satisfying
$$\forall n\in\mathbb{N},\quad \mathbb{P}(X_n=-1)=\frac{1}{2},\qquad \mathbb{P}(X_n=4n)=\frac{1}{8n},\qquad \mathbb{P}(X_n=0)=\frac{1}{2}-\frac{1}{8n},$$
and denote $S_n:=\sum_{k=1}^n X_k$. Prove that almost surely
$$\liminf_{n\to\infty}\frac{S_n}{n}<0<\limsup_{n\to\infty}\frac{S_n}{n}.$$

Problem 5 (qiuzhen-qualify 2025-fall)
Section: Probability & Statistics

(10 points) Let $B_t$ be a one-dimensional (1D) standard Brownian motion starting from 0. Define $X_t=\exp\big(B_t-\tfrac{1}{2}t\big)$.

(a) Prove that $X_t$ converges almost surely as $t\to\infty$.

(b) For any $t>0$, calculate $$\mathbb{E}\int_0^t X_s\,ds,\qquad \text{and}\qquad \operatorname{Var}\int_0^t X_s\,ds.$$

(c) Show that $$\int_0^{\infty} X_t\,dt<\infty\quad \text{a.s.}$$

Problem 6 (qiuzhen-qualify 2025-fall)
Section: Probability & Statistics

(15 points) Let $M$ be a $2\times2$ real symmetric random matrix of the form
$$M=\begin{pmatrix}X_1 & Y\\ Y & X_2\end{pmatrix},$$
where $X_1,X_2,$ and $Y$ are independent normal random variables: $X_1$ and $X_2$ follow the normal distribution $\mathcal{N}(0,2)$ with mean $0$ and variance $2$, and $Y$ follows the standard normal distribution $\mathcal{N}(0,1)$ with mean $0$ and variance $1$. Let $\lambda_1\ge \lambda_2$ denote the eigenvalues of $M$, and let $\mathbf{v}_1$ and $\mathbf{v}_2$ be the corresponding unit eigenvectors.

(a) Find the distribution of $\lambda_1+\lambda_2$.

(b) Find the distribution of $\lambda_1-\lambda_2$.

(c) Are $\lambda_1$ and $\lambda_2$ independent? Prove your claim.

(d) Prove that the joint distribution of $(\mathbf{v}_1,\mathbf{v}_2)$ is invariant under orthogonal transformations, i.e., for any $2\times2$ orthogonal matrix $O$, $(O\mathbf{v}_1,O\mathbf{v}_2)$ has the same distribution as $(\mathbf{v}_1,\mathbf{v}_2)$.

Problem 7 (qiuzhen-qualify 2025-fall)
Section: Probability & Statistics

(10 points) Suppose $\hat{\theta}_n$ is a real-valued consistent estimator of a parameter of interest $\theta$ based on $n$ observations, and it is further known that $\theta\in[-1,1]$. Let $T_n$ be a sufficient statistic for $\theta$. Show there exists a consistent estimator of $\theta$ based on $T_n$.

Problem 8 (qiuzhen-qualify 2025-fall)

(15 points) Let $X_1,\ldots,X_n$ be iid $\sim \exp(\lambda)$ (with pdf $\lambda e^{-\lambda x}$ for $x \ge 0$). Let $\phi = P_\lambda(X_1 > x) = e^{-\lambda x}$.

(a) Find a complete and sufficient statistic for $\lambda$.

(b) Find the UMVUE of $\phi$.

Problem 9 (qiuzhen-qualify 2025-fall)

(15 points) Let $X_1,\ldots,X_n$ be iid Bernoulli$(p)$ random variables with $0 < p < 1$.

(a) Let $g(p) = p^k + (1-p)^{n-k}$ where $k$ is a nonnegative integer and $0 \le k \le n$. Find the BUE (UMVUE) for $g(p)$.

(b) Prove that $\hat{X} = \frac{\sum_{i=1}^n x_i}{n}$ is an admissible estimator/rule of $p$ under the loss function
$$l(p,a) = \frac{(p-a)^2}{p(1-p)}.$$

Problem 10 (qiuzhen-qualify 2025-fall)

(10 points) Let $X_1,\ldots,X_n$ be iid from the uniform distribution $U(0, \theta)$ with $\theta > 0$ unknown. Suppose that the prior on $\theta$ is lognormal, i.e., $\ln \theta$ follows $N(\mu_0, \sigma_0^2)$ distribution where $\mu_0 \in \mathbb{R}$ and $\sigma_0 > 0$ are known constants.

(a) Find the posterior density of $\ln \theta$.

(b) Suppose that one defines the Bayes estimate for $\theta$ as the value that maximizes the posterior density of $\theta$. Find this Bayes estimator. Is it consistent for $\theta$? Please explain your answer.

Problem 11 (qiuzhen-qualify 2025-fall)

(10 points) Let $X_1,\ldots,X_n$ be iid random variables following a discrete uniform distribution on the set $1,2,\ldots,\theta$, where $\theta$ is an unknown positive integer. Let $\theta_0$ be a known positive integer. Let $X_{(n)}$ be the largest order statistic and $x_{(n)}$ be its realized value. We are interested in the hypothesis testing problems below at level $0 < \alpha < 1$.

(a) Consider testing $H_0 : \theta \le \theta_0$ versus $H_1 : \theta > \theta_0$. Show that
$$\phi(\mathbf{x}) = \begin{cases}
1, & \text{if } x_{(n)} > \theta_0, \\
\alpha, & \text{if } x_{(n)} \le \theta_0
\end{cases}$$
 is a (randomized) UMP test of size $\alpha$.

(b) Now consider testing $H_0 : \theta = \theta_0$ versus $H_1 : \theta \ne \theta_0$. Show that
$$\phi(\mathbf{x}) = \begin{cases}
1, & \text{if } x_{(n)} > \theta_0 \text{ or } x_{(n)} \le \theta_0 \alpha^{1/n}, \\
0, & \text{otherwise}
\end{cases}$$
 is a UMP test of size $\alpha$ when $\theta_0 \alpha^{1/n}$ is an integer.