13 problems

For $p,q>0$, let $\beta_{p,q}$ be the beta distribution on $[0,1]$ with density … A probability measure is freely infinitely divisible if it is an $n$-fold free additive convolutio…

Let $d\geq 2$. For $d+2$ independent random points in $\mathbb{R}^d$ drawn from the beta distribution with parameter $\beta>-1$, let $p_d(\beta)$ denote the probability that the po…

Let $\bbbeta=(\beta_1,\ldots,\beta_m)$ be the block-parameter vector for the block-beta random polytope $\mathcal{P}_{n,d}^{\bbbeta}$. Negative-parameter extension conjecture. The…

Monotonic beta convergence conjecture. The probability $\operatorname{\mathbb{P}}(W_n\leq p)$ tends monotonically to $1/2$ as $n$ tends to infinity, without imposing the alternativ…

Let $k\in\mathbb N_0$, let $\Re\mu>\mu_0+kq+1$, and let $\nu\in E_k$. Suppose that the beta distribution $\beta_{\mu,\nu}$ is a complex measure, meaning that…

Dual degenerate insertion conjecture. For every real $p<\frac{4}{\gamma^2}$,

Beta-mixture conjecture. In the general case, there exist non-negative constants $c_1,\dots,c_n$ with

Fix a positive real number $d4c1$ and define … where ${\mathfrak B}$ is beta distributed. Monotonicity conjecture. For real $d4e1,d4e1'$ satisfying … one has…

Let $d4e1,d4e2\ta$ be positive real numbers with $d4e1ta ecd4e2$. Let ${mathfrak B}$ be a beta-distributed random variable, and let $e{d4e1,d4e2}$ denote the associated equipoint.…

Jurek–Sato conjecture. $\beta_{a,b}^{-s}$ belongs to $\mathcal{G}$ for all $s\geq 1$.

Let $\operatorname{Beta}(\alpha_1,\alpha_2)$ denote a Beta random variable with parameters $\alpha_1,\alpha_2>0$, and let $s\ge1$. A positive random variable is generalized gamma c…

Let $\bm{\beta}_{p,q}$ and $\bm{\beta}'_{p,q}$ denote the beta and beta prime distributions, respectively, and let $\bm{\gamma}_p$ denote the gamma distribution, with parameters in…

Let $n\geqslant2$ and $\textbf{Y}=(Y_1,Y_2,\ldots,Y_n)$ be a vector of i.i.d. random variables. Let $G_a$ denote a gamma random variable with parameter $a$, and let $B_{a,b}$ denot…