79 problems

Let $\bm{z}$ denote a variable in the domain of a multivariate distribution, and let $f(\cdot)$ be any continuous function. Multivariate exponential distribution conjecture. In the…

Logarithmic convexity conjecture. For every $c\in\mathbb{R}$, $S_{n,c}$ is logarithmically convex; equivalently, $\log S_{n,c}$ is convex on $I_c$.

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

Let $\alpha\in(0,1)$ and $\gamma>0$, and let $\widehat h_{\alpha,\gamma}$ be the density on $(0,\infty)$ of the positive random variable $\widehat{\mathbf X}_{\alpha,\gamma}$, sati…

Let $\alpha\in(0,1)$ and $\gamma>0$, and let $\widehat{\mathbf X}_{\alpha,\gamma}$ be the positive random variable whose density $\widehat h_{\alpha,\gamma}$ solves … for $x>0$. A…

Cauchy-distribution conjecture. The Cauchy-distribution description of regular motions may hold more generally for integrable conservative systems.

Let $X$ and $Y$ be independent positive random variables, and define $U$ and $V$ by the transformation in the preceding independence property. Suppose that $U$ and $V$ are also ind…

Consider disordered systems whose couplings are sampled from a probability distribution, including the Bernoulli distribution as a discrete example. Discrete-distribution conjectur…

Nonexistence conjecture. The condition derived in the paper for the nonexistence of a Gerow–Robson density on $M(n)$ holds for every $n\geq 6$.

Conjecture on the eigenvalue sum. The sum of the eigenvalues is $0.75$. This proposed value concerns the eigenvalues associated with the Laplace distribution in the paper's orthogo…

Let $v>0$, $z>0$, and let $s$ satisfy … For the Lerch distribution, write $\mu$ for its mean and $\sigma^2$ for its variance. Lerch distribution overdispersion conjecture. Based on…

A one-dimensional stable distribution is a probability distribution on the real line whose characteristic function has the stable-law form. Unimodality conjecture. All one-dimensio…

Let $X_1,\ldots,X_n$ be independent and identically distributed random variables with an analytic density on $(0,\infty)$, and let $X_{(k;n)}$ denote the $k$-th order statistic. Ar…

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…

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…

Assume throughout that $calpha=4$ and $cbeta=2$. Let $p_n(t)=csum_{k=0}^n c_{n,k}t^k$ and define … For $n otincmathbb{N}$ and $1cleq kcleq n$, the coefficients $c{n,k}$ are those o…

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 $X_1 \sim \operatorname{Laplace}(0,1/\sqrt{2})$ and $X_2 \sim \operatorname{t}_5(0,\sqrt{3/5})$, as in the third example, and let $\mathcalB_0$ denote the solution set consider…

Let a dynamical system undergo an abrupt transition from a fixed-point regime to a chaotic regime, and consider its invariant measure at the boundary between these regimes. Beck's…

Manole et al.'s conjecture. A central limit theorem of this form should hold under the assumption

Strict lower-bound conjecture. For every $\kappa>1$,

Consider a family of smooth multivariate probability distributions. Multivariate monotonicity conjecture. It should be possible to formulate reasonable conditions under which there…

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 $x$, $y$, 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 $X$ 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 ……