19 problems

Let $X_1,\ldots,X_n$ be independent nonnegative random variables with $\mathbb E(X_i)=\mu_i\leq 1$ for each $i=1,\ldots,n$, and let $\delta>0$. Feige's conjecture. … This strengthe…

Burdzy–Pitman conjecture. For every $\delta\in(\frac{1}{2},1]$,

Let $Y$ be an absolutely continuous random variable, and let a linear combination of independent Bernoulli random variables be formed. Kuperberg's moment-matching conjecture. The m…

Let $n\geq 3$, and let $X_1,\dots,X_n$ be independent random variables with a common half-normal distribution. Write $\max\{X_1,\dots,X_n\}$ for their maximum. Arnold–Villasenor co…

Let $(R_i)_{i\geq 1}$ be the random variables considered in the paper, let $(a_n)$ satisfy $a_n\to\infty$, and let $(b_n)$ be a normalizing sequence. Normalized-maxima conjecture.…

Let $X$ and $Y$ be random variables, and suppose that the distribution $\mu_Y$ of $Y$ has non-trivial atoms: sets $A$ with $\mu_Y(A)>0$ such that every subset $A'\subset A$ satisfi…

Leader–Radcliffe's conjecture.

Critical-root growth conjecture. Under the assumptions of Theorem 3.1,

Let $U_1,\ldots,U_n$ be independent random variables uniformly distributed on $[-1,1]$, let $a=(a_1,\ldots,a_n)$ be a unit vector in $\mathbb{R}^n$, and let $h_1$ denote Shannon en…

Let $U_1,U_2,\ldots$ be independent random variables uniformly distributed on $[-1,1]$, let $p\in(0,1)$), and let $a=(a_1,\ldots,a_n)\in\mathbb{R}^n$. The sharp $L_p$-$L_2$ moment…

Optimal-strategy conjecture. After observing $I_{k}=1$, the player whose turn it is should give up his turn to his opponent if and only if $p_{i,k}<\frac{1}{2}$ for all $i>k$. This…

Let $n$ be the number of colours and let $m$ mutually independent fair dice each have $r$ sides, with no colour used more than once on the sides of any one die. Let $X$ be the numb…

Let $N\ge 3$, and let $l$ and $m$ be positive integers such that $1\le l\le N$, $2\le m\le N-1$, and both $l$ and $m$ are relatively prime to $N$. Let $X_l(m,N)$ be the random vari…

Uniform-boundedness conjecture. If $\{\xi_k\}$ are uniformly bounded random variables, for example if they have the uniform distribution on $[-1,1]$, then $\text{}$ should hold. Th…

Let $n \in \mathbb{Z}_{\ge 1}$, let $\mu_1 \ge \dotsc \ge \mu_n > 0$ satisfy … and let $X_1,\dotsc,X_n$ be independent non-negative random variables with $EX_i=1$ for every $i$. De…

Let $n \in \mathbb{Z}_{\ge 1}$, let $\mu_1 \ge \dotsc \ge \mu_n > 0$ satisfy … and let $X_1,\dotsc,X_n$ be independent non-negative random variables with $EX_i=1$ for every $i$. De…

Let $n\in\mathbb{N}$, let $\|\cdot\|_K$ be any norm on $\mathbb{R}^n$, and let $X_1,\dots,X_n$ be independent, non-compactly supported random variables on $\mathbb{R}_+$. For a vec…

Let $X$ and $Y$ be independent random variables on $\mathbb{R}_+$ that are not compactly supported, and let $\lambda\in(0,1)$. Weighted-average tail conjecture. … This conjecture i…

Let $\bX_1,\dots,\bX_n$ be jointly distributed $\{\pm1\}$-valued random variables. Conditioning on a random subset of $O(1/\eps^2)$ variables is known to reduce their average pairw…