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As noted in the recent answer by Yuval Peres, the sum of independent uniformly distributed random variables (r.v.'s) cannot have a normal distribution. The question is, what happens without the inde …

Let $a_1,\dots,a_n$ be real numbers such that $a_1^2+\dots+a_n^2=1$. Let $\eta_1,\dots,\eta_n$ be independent Rademacher random variables (r.v.'s), so that $P(\eta_i=\pm1)=\frac12$ for all $i$. Let $S …

$\newcommand{\Om}{\Omega}\newcommand{\F}{\mathcal F} $Let $X$ and $Y$ be random variables (r.v.'s) defined on a non-atomic probability space $(\Om,\F,P)$ such that $P(X<0)>0$ and $P(Y<0)>0$. Does then …

Yesterday the following question was asked by user sigmatau: I'm interested in the following question: given $n$ i.i.d. random variables $X_i \sim \mathcal{N}(0,\sigma^2_1), i=1,\ldots,n$ …

$\newcommand\Om\Omega$Let $(\Om,F,P)$ be a probability space. For some natural $n$, let $A_1,\dots,A_n$ be events, that is, members of the $\sigma$-algebra $F$. For $k\in[n]:=\{1,\dots,n\}$, these eve …

The following question was asked today: How many flips $n$ of a fair coin are needed to get at least one run of at least $k$ consecutive heads with probability $P_{k,n}\ge1/2$? The question w …

For $p\in(1,2)$, let $C_p$ be the smallest constant factor $C$ in the von Bahr–Esseen-type inequality \begin{equation}\label{eq:pair}\tag{1} E\Bigl\lvert\sum_{j=1}^n X_j\Bigr\rvert^p\le C\sum_{j=1 …

Let $Z_1,\dots,Z_n$ be iid random variables (r.v.'s) each uniformly distributed on $[0,1]$. Let $Z_{n:1}\le\cdots\le Z_{n:n}$ be the corresponding order statistics. For $i=1,\dots,n-1$, let $G_i:=Z_{n …

$\newcommand{\E}{\mathsf{E}}$ $\newcommand{\P}{\mathsf{P}}$ The following additional question was asked in a comment by user Oleg: Suppose that $(\Omega,\mathcal F,\P)$ is a probability space, $B …

$\newcommand{\al}{\alpha}$ For $i=1,\dots,n$, let \begin{equation} R_i:=\frac{X_i}{X_1+\dots+X_n}, \end{equation} where the $X_i$'s are iid standard exponential random variables. Let $$R_*:=\max_{1 …

Does there exist a probability distribution over the set $\{(x,y,z)\in[0,1]^3\colon x+y+z=3/2\}$ whose projection on each of the three coordinate axes is the uniform distribution over the interval $[ …

Let $p$ and $q$ be probability densities on $\mathbb R$, with respect to the Lebesgue measure $dx$. The corresponding Hellinger integral is $H(p,q):=\int_{\mathbb R}\sqrt{pq}\,dx$. Let now $p$ be t …

$\newcommand{\Z}{\mathbb{Z}} \newcommand{\J}{\mathcal{J}} \newcommand{\la}{\lambda} \newcommand{\1}{\mathbf{1}} \newcommand{\R}{\mathbb{R}}$ Take any $n\in[3;\infty]$. Here and in what follows, $[k;\ …

A question was recently asked by a new user, SomeoneHAHA, and then deleted by the user, after receiving an answer. I think the question and the answer (QA) to it may be of interest to some users. Ther …

Very recently, the following question was asked: Often, we encounder the assumption that $(\Omega,\mathcal{F},\mathbb{F},\mathbb{P})$ is a stochastic base on which a Brownian motion is defined. Ofte …