Related papers: Best lower bound on the probability of a binomial …

It is a well-known conjecture, sometimes attributed to Frankl, that for any family of sets which is closed under the union operation, there is some element which is contained in at least half of the sets. Gilmer was the first to prove a…

Let $S$ and $X$ be independent random variables, assuming values in the set of non-negative integers, and suppose further that both $\mathbb{E}(S)$ and $\mathbb{E}(X)$ are integers satisfying $\mathbb{E}(S)\ge \mathbb{E}(X)$. We establish a…

Let $B(n,p)$ denote a binomial random variable with parameters $n$ and $p$. Chv\'{a}tal's theorem says that for any fixed $n\geq 2$, as $m$ ranges over $\{0,\ldots,n\}$, the probability $q_m:=P(B(n,m/n)\leq m)$ is the smallest when $m$ is…

We provide a new lower bound on the length of the longest cycle of the binomial random graph $G(n,(1+\epsilon)/n)$ that holds w.h.p. for all $\epsilon=\epsilon(n)$ such that $\epsilon^3n\to \infty$. In the case $\epsilon\leq \epsilon_0$ for…

Let $G$ be a finite graph with minimum degree $r$. Form a random subgraph $G_p$ of $G$ by taking each edge of $G$ into $G_p$ independently and with probability $p$. We prove that for any constant $\epsilon>0$, if $p=\frac{1+\epsilon}{r}$,…

Let m_j denote the jth moment of a random variable X. One of the results is the inequality m_3 \leq c m_4^{3/4} with the best possible factor c over all random variables X with a non-positive mean.

We estimate from below the lower density of the set of prime numbers p such that p-1 has a prime factor of size at least p^c, where c lies in between 1/4 and 1/2. We also establish upper and lower bounds on the counting function of the set…

We prove that there is a constant $c >0$, such that whenever $p \ge n^{-c}$, with probability tending to 1 when $n$ goes to infinity, every maximum triangle-free subgraph of the random graph $G_{n,p}$ is bipartite. This answers a question…

Let $p$ be a fixed prime, and let $v(a)$ stand for the exponent of $p$ in the prime factorization of the integer $a$. Let $f$ and $g$ be two monic polynomials with integer coefficients and nonzero resultant $r$. Write $S$ for the maximum of…

We study the approximability of Max Ones when the number of variable occurrences is bounded by a constant. For conservative constraint languages (i.e., when the unary relations are included) we give a complete classification when the number…

For n>=1 let X_n be a vector of n independent Bernoulli random variables. We assume that X_n consists of M "blocks" such that the Bernoulli random variables in block i have success probability p_i. Here M does not depend on n and the size…

Consider the problem of maximizing the probability of stopping with one of the two highest values in a Bernoulli random walk with arbitrary parameter $p$ and finite time horizon $n$. Allaart \cite{Allaart} proved that the optimal strategy…

While useful probability bounds for $n$ pairwise independent Bernoulli random variables adding up to at least an integer $k$ have been proposed in the literature, none of these bounds are tight in general. In this paper, we provide several…

We consider Khintchine type inequalities on the $p$-th moments of vectors of $N$ pairwise independent Rademacher random variables. We establish that an analogue of Khintchine's inequality cannot hold in this setting with a constant that is…

We consider the Last-Success-Problem with $n$ independent Bernoulli random variables with parameters $p_i>0$. We improve the lower bound provided by F.T. Bruss for the probability of winning and provide an alternative proof to the one given…

We address the problem of bounding below the probability of error under maximum likelihood decoding of a binary code with a known distance distribution used on a binary symmetric channel. An improved upper bound is given for the maximum…

Let $a_1, \dots, a_n \in \mathbb{R}$ satisfy $\sum_i a_i^2 = 1$, and let $\varepsilon_1, \ldots, \varepsilon_n$ be uniformly random $\pm 1$ signs and $X = \sum_{i=1}^{n} a_i \varepsilon_i$. It is conjectured that $X = \sum_{i=1}^{n} a_i…

For frequentist settings in which parameter randomness represents variability rather than uncertainty, the ideal measure of the support for one hypothesis over another is the difference in the posterior and prior log odds. For situations in…

Given a set of independent Poisson random variables with common mean, we study the distribution of their maximum and obtain an accurate asymptotic formula to locate the most probable value of the maximum. We verify our analytic results with…

It is well known that a binomial $(n,p)$ can be approximated by a Poisson distribution with parameter $np$. The typical approach in undergraduate probability texts is to show a convergence result for the distribution of the binomial as $n$…