24 problems

Let $calpha>2$, and for each $n$ consider the coefficient sequence $(c_{n,k})_{k\in\{1,2,\ldots,n\}}$. Unimodality conjecture. For every choice of $calpha>2$, the sequence … is uni…

Bounded output-density conjecture. For every such $\mathcal{D}$, the probability density function of $\mathcal{D}^{\circ}$ is bounded from above by a constant $C$ that depends only…

Let $\mathbb{R}_+^d$ denote the positive orthant, and let MPH$^\ast$ distributions have general phase-type (PH) margins. The mPH minimality conjecture. The mPH class is the smalles…

Let $X\sim f_X$ and $Z\sim f_Z$ be independent random variables, where $f_X$ and $f_Z$ are densities with cumulative distribution functions $F_X$ and $F_Z$, respectively. Integral…

Let $X$ and $Z$ be independent draws from arbitrary unknown probability distributions with densities $f_X$ and $f_Z$, satisfying … Let $Y$ be drawn from the mixture distribution ……

Let $calpha>2$ and let $cbeta$ be the parameter defining the coefficient arrays $(c_{n,k})$ and $(b_{n,k})$. Uniform approximation conjecture. If … then … as $n\to\infty$. Such an…

Let $x$, $y$, and $z$ be three points on the real line, where $x$ and $z$ are generated independently from two distinct distributions and $y$ is generated from their 50--50 mixture…

Let $d_1\in\mathbb{N}$ and $d_2\in\mathbb{N}$ with $d_2\geq 5$. Let $X_{d_1,d_2}$ be an $F$-random variable with numerator and denominator degrees of freedom $d_1$ and $d_2$, respe…

Gaunt–Merkle median bounds conjecture. For $r>0$,

Consider a random multidigraph on $n$ vertices in which the outdegree of each vertex is independently distributed as $\operatorname{Ge}(1-p)$, with $p\in(0,1)$, and endpoints of ou…

Generalized hyperbolic median conjecture. The median satisfies

McKay Type I median conjecture. The function $\phi\mapsto\operatorname{Med}(Z_{m,c,\phi})$ is strictly increasing for $c\in(1,\infty)$. Consequently,

Variance-gamma median conjecture. The function $\sigma\mapsto\operatorname{Med}(V_{r,\theta,\sigma})$ is strictly decreasing for $\sigma\in(0,\infty)$, and consequently

Gamma-sum median conjecture. The function $\alpha\mapsto\operatorname{Med}(Z_\alpha)$ is non-decreasing for $\alpha\in(0,1)$. This is one of two conjectures concerning medians of s…

Let $\kappa<8$, and consider the backward $SLE_{\kappa}$ flow and its $SLE_{\kappa}$ trace. At any fixed deterministic time $t>0$, let the tip of the trace be viewed through the co…

Let $D_n$ be the total number of defaults among $n$ banks, and let $p$ be the corresponding asymptotic default probability. Default-count conjecture. The random variable $D_n$ shou…

Let the log-return distributions of large-cap equities be viewed as probability distributions, with scale and location allowed to vary. First Universality Conjecture. The distribut…

Let the unit square be equipped with the uniform probability distribution, and let an optimal set of $n$-means have associated quantizer points. Conjecture on optimal quantizers fo…

Let the unit disc be equipped with the uniform probability distribution. For an optimal set of $n$-means, consider the associated Voronoi regions. Conjecture on optimal quantizers…

Let $G(n,\pi)$ be the unit disk graph, and let two vertices be displaced by $r_{xy}$. Write $\sigma_{r_{xy}}$ for the number of geodesic paths between them. Negative-binomial conje…

Dirichlet-weighted Arcsin conjecture. The random variable $S_n$ has the power semicircle distribution on $(-a,a)$ with $\lambda=\frac{n-1}{2}$ and density

Inverse-chi-square conjecture. Then

Cauchy ratio conjecture. The random variable

Monomial distribution conjecture. The Wald variable satisfies