7 problems

Samuels' conjecture.

For a coupon collector model with $N$ coupon types and positive probability vector $p=(p_1,\ldots,p_N)$, let $U_j^N$ be the number of coupon types that have appeared fewer than $j$…

For each sufficiently large integer $n$, let $v_1,\ldots,v_n$ be arbitrary unit vectors in $\mathbb{R}^2$, and let $\epsilon_1,\ldots,\epsilon_n$ be independent Rademacher random v…

Signs-model conjecture. Among models in $\mathcal{D}_{\geq p}(Q_3)$, the signs model minimises the probability that $Q_3$ is connected.

Let $X_1,\ldots,X_n$ be iid random variables supported on $[0,1]$, with mean $m$, and let an extremal distribution mean one attaining … A distribution is binary if it has two suppo…

Let $X_1,\ldots,X_n$ be iid random variables supported on $[0,1]$, with mean $m$ satisfying $0<m<1$, and write $S_n=X_1+\cdots+X_n$. Define … For $n\geq2$ and $0\leq t<mn$, call a…

Two-error-size conjecture. For each $m$ and $t$, there is an extremal population with just two error sizes, one of which is $0$ or $1$.