Problem 1 (yau-contest 2012)
Section: Probability and Statistics

Solve the following two problems:
1) An urn contains $b$ black balls and $r$ red balls. One of the balls was drawn at random, and putted back in the urn with $a$ additional balls of the same color. Now suppose that the second ball drawn at random is red. What is the probability that the first ball drawn was black?
2) Let $(X_n)$ be a sequence of random variables satisfying $\lim_{a\to\infty}\sup_{n\ge1} P(|X_n|>a)=0$. Assume that a sequence of random variables $(Y_n)$ converges to $0$ in probability. Prove that $(X_n Y_n)$ converges to $0$ in probability.

Problem 2 (yau-contest 2012)
Section: Probability and Statistics

Solve the following two problems:
1) Let $(\Omega,\mathcal{F},P)$ be a probability space, $\mathcal{G}$ be a sub-algebra of $\mathcal{F}$. Assume that $X$ is a non-negative integrable random variable. Set $Y=E[X\mid\mathcal{G}]$. Prove that
(a) $\{X>0\}\subset\{Y>0\}$, a.s.;
(b) $\{Y>0\}=\mathrm{ess.inf}\{A: A\in\mathcal{G},\ [X>0]\subset A\}$.
2) 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 $E(X\mid X+Y)$.

Problem 3 (yau-contest 2012)
Section: Probability and Statistics

Suppose that $\{p(i,j): i=1,2,\ldots,m;\ j=1,2,\ldots,n\}$ is a finite bivariate joint probability distribution, that is,
$$p(i,j)>0,\qquad \sum_{i=1}^m\sum_{j=1}^n p(i,j)=1.$$
(i) Can $\{p(i,j)\}$ be always expressed as
$$p(i,j)=\sum_k \lambda_k a_k(i)b_k(j)$$
for some finite $\lambda_k\ge0$, $\sum_k\lambda_k=1$, $a_k(i)\ge0$, $\sum_{i=1}^m a_k(i)=1$, $b_k(j)\ge0$, $\sum_{j=1}^n b_k(j)=1$?
(ii) Prove or disprove the above relation by use of conditional probability.

Problem 4 (yau-contest 2012)
Section: Probability and Statistics

Let $X_1,\cdots,X_m$ be an independent and identically distributed (i.i.d.) random sample from a cumulative distribution function (CDF) $F$, and $Y_1,\cdots,Y_n$ an i.i.d. random sample from a CDF $G$. We want to test $H_0:F=G$ versus $H_1:F\ne G$. The total sample size is $N=m+n$. Consider the following two nonparametric tests:
• The Wilcoxon rank sum tests. The test proceeds by first ranking the pooled $X$ and $Y$ samples and then taking the sum of the ranks associated with the $Y$ sample. Let $R_{y_1},\cdots,R_{y_n}$ be the rankings of the sample $y_1<\cdots<y_n$ from the pooled sample in increasing order. The Wilcoxon rank sum statistic is defined as $W=\sum_{j=1}^n R_{y_j}$.
• The Mann-Whitney $U$-test. Let $U_{ij}=1$ if $X_i<Y_j$, and $U_{ij}=0$ otherwise. The Mann-Whitney $U$-statistic is defined as $U=\sum_{i=1}^m\sum_{j=1}^n U_{ij}$. The probability $\gamma=P(X<Y)$ can be estimated as $U/(mn)$. The decision rule is based on assessing if $\gamma=0.5$.
Assume that there are no tied data values.
(a) Show that $W=U+\tfrac{1}{2}n(n+1)$, which shows that the two test statistics differ only by a constant and yield exactly the same $p$-values.
(b) Using the central limit theorem, the Wilcoxon rank sum statistic $W$ can be converted to a $Z$-variable, which provides an easy-to-use approximation. The transformation is
$$Z_W=\frac{W-\mu_W}{\sigma_W},$$
where $\mu_W$ and $\sigma_W^2$ are the mean and variance of $W$ under $H_0$. Show that $\mu_W=\tfrac{1}{2}n(N+1)$ and $\sigma_W^2=\tfrac{1}{12}mn(N+1)$.

Problem 5 (yau-contest 2012)
Section: Probability and Statistics

Let $X$ be a random variable with $EX^2<\infty$, and $Y=|X|$. Assume that $X$ has a Lebesgue density symmetric about $0$. Show that random variables $X$ and $Y$ are uncorrelated, but they are not independent.

Problem 6 (yau-contest 2012)
Section: Probability and Statistics

Let $E_1,\cdots,E_n$ be i.i.d. random variables with $E_i\sim \mathrm{Exponential}(1)$. Let $U_1,\cdots,U_n$ be i.i.d. uniformly (on $[0,1]$) distributed random variables. Further, assume that $E_1,\cdots,E_n$, and $U_1,\cdots,U_n$ are independent.
(a) Find the density of $X=(E_1+\cdots+E_m)/(E_1+\cdots+E_n)$, where $m<n$.
(b) Show that $Y=\frac{(n-m)X}{m(1-X)}$ is distributed as the $F$-distribution with degrees of freedom $(2m,2(n-m))$.
(c) Find the density of $(U_1\cdots U_n)^{-X}$.