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

(a) Let $X$ and $Y$ be two random variables with zero means, variance $1$, and correlation $\rho$. Prove that
$$\mathbb{E}[\max\{X^2, Y^2\}] \le 1 + \sqrt{1-\rho^2}.$$

(b) Let $X$ and $Y$ have a bivariate normal distribution with zero means, variances $\sigma^2$ and $\tau^2$, respectively, and correlation $\rho$. Find the conditional expectation $\mathbb{E}(X\mid Y)$.

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

We flip a fair coin until heads turns out twice consecutively. What is the expected number of flips?

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

Let $(X_n, n \ge 1)$ be a sequence of independent Gaussian variables, with respective mean $\mu_n$, and variance $\sigma_n^2$.

(a) Prove that if $\sum_n X_n^2$ converges in $L^1$, then $\sum_n X_n^2$ converges in $L^p$, for every $p \in [1, \infty)$.

(b) Assume that $\mu_n = 0$, for every $n$. Prove that if $\sum_n \sigma_n^2 = \infty$, then
$$\mathbb{P}\Big(\sum_n X_n^2 = \infty\Big) = 1.$$

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

Let $X_1,\ldots,X_n$ be a random sample of size $n$ from the exponential distribution with pdf $f(x;\theta) = \theta^{-1} \exp(-x/\theta)$ for $x, \theta > 0$, zero elsewhere. Let $Y_1 = \min\{X_1,\ldots,X_n\}$. Consider an estimator $nY_1$.

(a) Show this estimate is unbiased.

(b) Prove or disprove: This estimate is a consistent estimator.

(c) Prove or disprove: This estimate is an efficient estimator.

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

Let the independent normal random variables $Y_1,\ldots,Y_n$ have, respectively, the probability density functions $N(\mu, \gamma^2 x_i^2)$, $i=1,\ldots,n$, where the given $x_1,\ldots,x_n$ are not all equal and no one of which is zero.

(a) Construct a confidence interval for $\gamma$ with significance level $1-\alpha$.

(b) Discuss the test of the hypothesis $H_0 : \gamma = 1,\ \mu$ unspecified, against all alternatives $H_1 : \gamma \ne 1,\ \mu$ unspecified.