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Let $X$ be a discrete random variable supported on $\{−5,\dots, 6\}$ in which the outcomes have the following respective probabilities: $$\begin{array}{rc} n & p(n) \\ \hline −5 & 6/36 \\ −4 & 0 \\ −3 …

Suppose we have a probability distribution $p(\cdot)$ supported on the integers between $-m$ and $m$ for some positive integer $m$, with $\sum_k kp(k) = 0$. Suppose furthermore that all $p(k)$ are rat …

Part of what I do is study typical behavior of large combinatorial structures by looking at pseudorandom instances. But many commercially available pseudorandom number generators have known defects, …

Consider two independent copies of IID random walk on ${\bf Z}$ starting from $0$, and let $N_1(t)$ (resp. $N_2(t)$) denote the number of times, up to time $t$, that the first (resp. second) walker ha …

What are the feasible $2^n$-tuples of entropies $h(S) := H(X_{i_1},\dots,X_{i_{|S|}})$ where $X_1,\dots,X_n$ are discrete random variables with some (unknown) joint probability distribution as $S=\{i_ …

Starting from the length-1 list whose only entry is 1, iterate the process of replacing the last (and largest) entry in the list of length $n$ (call that entry $m$) by the two numbers $mU_n$ and $m(1- …

Suppose we have a discrete group $G$ acting on a compact set $X \subseteq \mathbb{R}^d$ via measure-preserving homeomorphisms, and suppose we have a point $x$ whose orbit $Gx$ is finite (say $|Gx| = n …

Given a function $f$ from a finite set $X$ to itself, it seems natural to consider $\kappa_f := (\sum_{x \in X} |f^{-1}(x)|^2)/|X|$ as a measure of the non-invertibility of $f$: it equals 1 if $f$ is …

Suppose $X_1,\dots,X_n$ are jointly distributed random variables such that the random $n$-tuple $(X_1,\dots,X_n)$ is uniformly distributed on the set of $n$-tuples of nonnegative integers summing to $ …

More than one combinator(ial?)ist has asked me to recommend a good book to learn probability from, and I never know what to say; the probability theory that I use in my research up was mostly learned …

The $2m$th moment of the (random) area of the triangle whose vertices are three independent, uniformly distributed random points on the unit circle appears to be $((3m)!/(m!)^3)/16^m$. Can anyone prov …

Given a connected graph $G$ with a connected subgraph $H$, we can consider the uniform distribution on the set of all sequences $H_0, H_1, \dots, H_r$ where $r = |E(G) \setminus E(H)|$, $H_0 = H$, $H_ …

Let $X_1,X_2,\dots$ be iid draws from the uniform distribution on $\{1,2,...,m\}$, and let the random variable $N$ be the minimum $j$ such that $X_j = X_i$ for some $i<j$. I'm aware that the expected …

Let $p(x) = \sum_{k \geq 0} a_k x^k$ where the $a_k$'s are IID random variables taken from a mean-zero random variable taking finitely many values in $\mathbb{R}$; it clearly converges for $-1<x<1$. I …

Does anyone know of any work on the following model or variants thereof?: Finitely many chips are distributed on the integers at time 0. To find the distribution at time $t+1$, take all the chips at …