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Consider the following recurrence relation in two variables: $$f(a, b) = \frac{a}{a+b} f(a-1,b)+ \frac{b}{a+b}f(a+1,b-1) $$ for positive integers $a$ and $b$, with the boundary conditions $f(0,b)=0$ f …

Consider the integers $\{1,\dots, N\}$ for some positive integer $N$. Let us suppose that for each $\{1, \dots, N\}$ there is an associated probability $p_1, \dots, p_N$. We also define an integer th …

Say we toss $d$ $k$-wise independent coins, each with probability $1/d$ of getting a head. What is the highest lower bound one can give for the probability of getting exactly one head? In the first i …

Say we toss $d$ pairwise independent coins, each with probability $1/d$ of getting a head. What is the highest lower bound one can give for the probability of getting exactly one head? If they had bee …

If $X_i$ is a sequence of $d$ dimensional i.i.d. integer valued random vectors with covariance matrix $\Sigma$ and $\mathbb{E}(X_i) = \mu$. Let each element of $X_i$ be chosen i.u.d. from $\{-1,1\}$. …

In a recent math.se question the following was asked which I have slightly edited. " Consider a fixed and given $n$ by $n$ matrix $M$ whose elements are chosen from $\{-1,1\}$. Consider also a random …

I am interested in the following simple looking problem on which I am stuck. Let $M$ be a fixed $m$ by $n$ matrix with $\pm1$ elements. Let $x$ and $y$ be two independently sampled random $n$-dimen …

Let $A$ be a random $m$ by $n$ rectangular sign matrix, chosen uniformly at random, with $m < n$. Let $B = A^T A$. We know, for example, that $B$ is a square and symmetric $n$ by $n$ matrix with all …

Consider a random $m$ by $n$ matrix $M$ with $m \leq n$, chosen uniformly over all those whose elements are in $\{0,1\}$ (or $\{-1,1\}$ if it is any easier). Is there any mathematical theory that …

[Question edited and changed a little on June 14 2015] Consider an $n$-dimensional vector $v$ with $v_i \in \{-1,1\}$. Now consider an $n$-dimensional vector $w$ with $w_i \in \{-1,0,1\}$. The ele …

The following problems come from something I worked on (with my coauthors) related to proving a new time lower bound for streaming problems. Having worked on these problems for some time with little …

Throw $n$ balls into $n$ bins, and let $X_n$ be the max load. That is the number of balls in the fullest bin. It is known that if the balls are thrown uniformly and independently at random then $\mat …

Imagine you sample $n$ numbers with replacement uniformly from the integers $1,\dots, n$ (we can assume $n$ is large). Let $X$ be the minimum of these samples. I am interested in $\mathbb{E}(X)$ but …

Imagine you sample $n$ number with replacement uniformly from the integers $1,\dots, n$. Let $X$ be the minimum of these samples. I am interested in $\mathbb{E}(X)$ but with a twist. All I know is t …

Consider $d$ dimensional space cut by $n$ hyperplanes in general position, each one of which goes through the origin. The number of distinct regions created is known to be: $$2\sum_{i=0}^{d-1} {n -1 …