12 problems

Let $k\ge2$ be fixed, let $\lambda\in(\frac{1}{k}\ln2,1)$, and write $\kappa=\frac{k(k+1)}{2}$. Let $\nu_k(\lambda)$ denote the median of the Poisson distribution of order $k$. Con…

Asymptotic median-difference conjecture.

Median offset limit conjecture.

Median offset monotonicity conjecture. The function $\alpha_{k,n}-n$ is decreasing in $n$ under one of the following conditions: (i) $k=2$ and $n\ge3\kappa$; (ii) $k\in[3,6]$ and…

Median offset bound conjecture. For fixed $k>1$ and $n\ge\kappa$,

Exactness conjecture for the median asymptotics. The terms in these two formulas that depend only on $k$ are exact.

Poisson order-$k$ median conjecture. Numerical evidence suggests that, for all $k\ge1$ and $n\ge\kappa$,

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-difference median conjecture. The function $\alpha\mapsto\operatorname{Med}(Z_\alpha)$ is non-increasing for $\alpha\in(2,\infty)$. The source notes that, if the preceding co…

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…