The random variables $X, Y$ can each take a finite number of integer values. They are not necessarily independent. Express $P(\min(X,Y)=k)$ in terms of $p_1=P(X=k)$, $p_2=P(Y=k)$ and $p_3=P(\max(X,Y)=k)$.

For problem 11 , i couldn’t find the correct translation , so i just posted the hungarian version . If anyone could translate it ,i would be very thankful . [tip=see hungarian]Az $X$ ́es$ Y$ valo ́s ́ert ́eku ̋ v ́eletlen v ́altoz ́ok maxim ́alkorrel ́acio ́ja az $f(X)$ ́es $g(Y )$ v ́altoz ́ok korrela ́cio ́j ́anak szupr ́emuma az olyan $f$ ́es $g$ Borel m ́erheto ̋, $\mathbb{R} \to \mathbb{R}$ fu ̈ggv ́enyeken, amelyekre $f(X)$ ́es $g(Y)$ v ́eges sz ́ora ́su ́. Legyen U a $[0,2\pi]$ interval- lumon egyenletes eloszl ́asu ́ val ́osz ́ınu ̋s ́egi v ́altozo ́, valamint n ́es m pozit ́ıv eg ́eszek. Sz ́am ́ıtsuk ki $\sin(nU)$ ́es $\sin(mU)$ maxim ́alkorrela ́ci ́oja ́t. [/tip] Edit: [hide=Translation thanks to @tintarn] The maximal correlation of two random variables $X$ and $Y$ is defined to be the supremum of the correlations of $f(X)$ and $g(Y)$ where $f,g:\mathbb{R} \to \mathbb{R}$ are measurable functions such that $f(X)$ and $g(Y)$ is (almost surely?) finite. Let $U$ be the uniformly distributed random variable on $[0,2\pi]$ and let $m,n$ be positive integers. Compute the maximal correlation of $\sin(nU)$ and $\sin(mU)$. (Remark: It seems that to make sense we should require that $E[f(X)]$ and $E[g(Y)]$ as well as $E[f(X)^2]$ and $E[g(Y)^2]$ are finite. In fact, we may then w.l.o.g. assume that $E[f(X)]=E[g(Y)]=0$ and $E[f(Y)^2]=E[g(Y)^2]=1$.)[/hide]

$x$ and $y$ are chosen at random (with uniform density) from the interval $(0, 1)$. What is the probability that the closest integer to $x/y$ is even?

Let $\xi_{(k_1, k_2)}, k_1, k_2 \in\mathbb N$ be random variables uniformly bounded. Let $c_l, l\in\mathbb N$ be a positive real strictly increasing infinite sequence such that $c_{l+1}/ c_l$ is bounded. Let $d_l=\log \left(c_{l+1}/c_l\right), l\in\mathbb N$ and suppose that $D_n=\sum_{l=1}^n d_l\uparrow \infty$ when $n\to\infty$ Suppose there exist $C>0$ and $\varepsilon>0$ such that $$\left| \mathbb E \left\{ \xi_{(k_1,k_2)}\xi_{(l_1,l_2)}\right\}\right| \leq C\prod_{i=1}^2 \left\{ \log_+\log_+\left( \frac{c_{\max\{ k_i, l_i\}}}{c_{\min\{ k_i, l_i\}}}\right)\right\}^{-(1+\varepsilon)}$$ for each $(k_1, k_2), (l_1,l_2)\in\mathbb N^2$ ($\log_+$ is the positive part of the natural logarithm). Show that $$\lim_{\substack{n_1\to\infty \\ n_2\to\infty}} \frac{1}{D_{n_1}D_{n_2}}\sum_{k_1=1}^{n_1} \sum_{k_2=1}^{n_2} d_{k_1}d_{k_2}\xi_{(k_1,k_2)}=0$$ almost surely. (translated by j___d)

Let the sequence of random variables $ \{ X_m, \; m \geq 0\ \}, \; X_0=0$, be an infinite random walk on the set of nonnegative integers with transition probabilities \[ p_i=P(X_{m+1}=i+1 \mid X_m=i) >0, \; i \geq 0 \,\] \[ q_i=P(X_{m+1}=i-1 \mid X_m=i ) >0, \; i>0.\] Prove that for arbitrary $ k >0$ there is an $ \alpha_k > 1$ such that \[ P_n(k)=P \left ( \max_{0 \leq j \leq n} X_j =k \right)\] satisfies the limit relation \[ \lim_{L \rightarrow \infty} \frac 1L \sum_{n=1}^L P_n(k) \alpha_k ^n < \infty.\] [i]J. Tomko[/i]

We throw $ N$ balls into $ n$ urns, one by one, independently and uniformly. Let $ X_i\equal{}X_i(N,n)$ be the total number of balls in the $ i$th urn. Consider the random variable \[ y(N,n)\equal{}\min_{1 \leq i \leq n}|X_i\minus{}\frac Nn|.\] Verify the following three statements: (a) If $ n \rightarrow \infty$ and $ N/n^3 \rightarrow \infty$, then \[ P \left(\frac{y(N,n)}{\frac 1n \sqrt{\frac Nn}}<x \right) \rightarrow 1\minus{}e^{\minus{}x\sqrt{2/ \pi}} \;\textrm{for all}\ \; x>0 \ .\] (b) If $ n\rightarrow \infty$ and $ N/n^3 \leq K$ ($ K$ constant), then for any $ \varepsilon > 0$ there is an $ A > 0$ such that \[ P(y(N,n) < A) > 1\minus{}\varepsilon .\] (c) If $ n \rightarrow \infty$ and $ N/n^3 \rightarrow 0$ then \[ P(y(N,n) < 1) \rightarrow 1.\] [i]P. Revesz[/i]

Let $\{U_{n,1},...,U_{n,n}\}_{n=1}^\infty$ be iid rv, uniformly distributed over [0,1] , and for $\alpha\geq 1$ consider the sets $\{[n^\alpha U_{n,1}],...,[n^\alpha U_{n,n}]\}$ , where [·] denotes the whole part. Prove that the elements of the sets $H_n\cap(\cup_{m=n+1}^\infty H_m)$ form an almost surely bounded sequence if and only if $\alpha>3$.

Let $ A_1,...,A_n$ be arbitrary events in a probability field. Denote by $ C_k$ the event that at least $ k$ of $ A_1,...,A_n$ occur. Prove that \[ \prod_{k=1}^n P(C_k) \leq \prod_{k=1}^n P(A_k).\] [i]A. Renyi[/i]

Let $ Z_1,\,Z_2\dots,\,Z_n$ be $ d$-dimensional independent random (column) vectors with standard normal distribution, $ n \minus{} 1 > d$. Furthermore let \[ \overline Z \equal{} \frac {1}{n}\sum_{i \equal{} 1}^n Z_i,\quad S_n \equal{} \frac {1}{n \minus{} 1}\sum_{i \equal{} 1}^n(Z_i \minus{} \overline Z)(Z_i \minus{} \overline Z)^\top\] be the sample mean and corrected empirical covariance matrix. Consider the standardized samples $ Y_i \equal{} S_n^{ \minus{} 1/2}(Z_i \minus{} \overline Z)$, $ i \equal{} 1,2,\dots,n$. Show that \[ \frac {E|Y_1 \minus{} Y_2|}{E|Z_1 \minus{} Z_2|} > 1,\] and that the ratio does not depend on $ d$, only on $ n$.

Let $n\geq 3$ be integer. Given two pairs of $n$ cards numbered from 1 to $n$. Mix the $2n$ cards up and take the card 3 times every one card. Denote $X_1,\ X_2,\ X_3$ the numbers of the cards taken out in this order taken the cards. Find the probabilty such that $X_1<X_2<X_3$. Note that once a card taken out, it is not taken a back.

Six cities $A, B, C, D, E$, and $F$ are located on the vertices of a regular hexagon in that order. $G$ is the center of the hexagon. The sides of the hexagon are the roads connecting these cities. Further more, there are roads connecting cities $B, C, E, F$ and $G$, respectively. Because of raining, one or more roads maybe destroyed. The probability of the road keeping undestroyed between two consecutive cities is $p$. Determine the probability of the road between cities $A$ and $D$ is undestroyed.

There are ${n}$ tokens in a pack. Some of them (at least one, but not all) are white and the rest are black. All tokens are extracted randomly from the pack, one by one, without putting them back. Let ${X_i}$ be the ratio of white tokens in the pack before the ${i^{\text{th}}}$ extraction and let \[ \displaystyle T =\max \{ |X_i-X_j| : 1 \leq i \leq j \leq n\}.\] Prove that ${\Bbb{E}(T) \leq H(\Bbb{E}(X_1))},$ where ${H(x)=-x\ln x -(1-x)\ln(1-x)}.$ [i]Proposed by Tamás Móri[/i]

Assign independent standard normally distributed random variables to the vertices of an n-dimensional cube. Say one vertex is greater than another if the assigned number is greater. Define a random walk on the vertices according to the following rules: a) the starting point is chosen from all the vertices with equal probability, b) during our journey, if we reach a vertex such that there are adjacent vertices which have higher values, we choose the next vertex with equal probability, c) if there is none, we stop. Prove that $\forall\varepsilon>0 \,\exists K\, \forall n>1$ $$P(\lambda> K \log n) <\varepsilon$$ where $\lambda$ is the number of steps of the random walk.

Let $x_1,x_2,\cdots,x_n$ be iid rv. $S_n=\sum x_k$ (a) let $P(|x_1|\leq 1)=1$ , $E[x_1]=0$ , $E[x_1^2]=\sigma^2>0$ Prove that $\exists C>0$ , $\forall u\geq 2n\sigma^2$ $P(S_n\geq u)\leq e^{-C u \log(u/n\sigma^2)}$ (b) let $P(x_1=1)=P(x_1=-1)=\sigma^2/2$ , $P(x_1=0)=1-\sigma^2$ Prove that $\exists B_1<1,B_2>1,B_3>0$ , $\forall u\geq1, B_1 n\geq u\geq B_2 n\sigma^2$ $P(S_n\geq u)>e^{-B_3 u \log(u/n\sigma^2)}$

Let $ A$ and $ B$ be nonsingular matrices of order $ p$, and let $ \xi$ and $ \eta$ be independent random vectors of dimension $ p$. Show that if $ \xi,\eta$ and $ \xi A\plus{} \eta B$ have the same distribution, if their first and second moments exist, and if their covariance matrix is the identity matrix, then these random vectors are normally distributed. [i]B. Gyires[/i]

Let $ \xi_1,\xi_2,...$ be independent, identically distributed random variables with distribution \[ P(\xi_1=-1)=P(\xi_1=1)=\frac 12 .\] Write $ S_n=\xi_1+\xi_2+...+\xi_n \;(n=1,2,...),\ \;S_0=0\ ,$ and \[ T_n= \frac{1}{\sqrt{n}} \max _{ 0 \leq k \leq n}S_k .\] Prove that $ \liminf_{n \rightarrow \infty} (\log n)T_n=0$ with probability one. [i]P. Revesz[/i]