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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 …

This probability problem seems interesting and I don't know if it has been solved before. If you pick $n$ points uniformly at random from the surface of a $d$ dimensional sphere of radius $r>1$ with …

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 …

Consider a degree $n$ polynomial $P(x)$ with coefficients $c_i \in \{-1,0,1\}$ chosen uniformly and independently. What is the probability that $P(x)$ has a root which is a root of unity? Prev …

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 …

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 …

[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 …

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 …

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 …

Consider Toeplitz matrices where the entries in the first row and column (which define the whole matrix) are independently chosen to be either $1$ or $0$ with probability $1/2$. Define $p_n$ to be the …

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 …

If you toss a coin $2\ell-1$ times you get a sequence of outcomes, say, $HTHTHTH$ for $\ell = 4$. I am trying to work out the probability that there are at least two runs (in other words contiguous s …

If you sample $n$ vectors each with $m$ entries, with each entry chosen from the set $\{-1, 1\}$, how can you calculate the expected maximum absolute value of the inner product between all pairs of ve …

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 …

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 …