Related papers: A study on the negative binomial distribution moti…

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…

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…

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 $\{0,1,\ldots,n\}$, the probability $q_m:=P(B(n,m/n)\leq m)$ is the smallest when $m$ is…

Let $B(n,p)$ denote a binomial random variable with parameters $n$ and $p$. Vas\v{e}k Chv\'{a}tal conjectured 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…

Consider the probability that a binomial random variable Bi$(n,m/n)$ with integer expectation $m$ is at most its expectation. Chv\'atal conjectured that for any given $n$, this probability is smallest when $m$ is the integer closest to…

Let $X_{d_1, d_2}$ be an $F$-random variable with parameters $d_1$ and $d_2,$ and expectation $E[X_{d_1, d_2}]$. In this paper, for any $\kappa>0,$ we investigate the infimum value of the probability $P(X_{d_1, d_2}\leq \kappa E[X_{d_1,…

In a recent paper, Svante Janson has considered a conjecture suggested by Va\v{s}ek Chv\a'atal dealing with the probability that a binomial random variable with parameters $n$ and $m/n$ - where $m$ is an integer - exceeds its expectation…

Let $X$ be a random variable distributed according to the binomial distribution with parameters $n$ and $p$. It is shown that $P(X>EX)\ge1/4$ if $1>p\ge c/n$, where $c:=\ln(4/3)$, the best possible constant factor.

Let $\{X_\alpha\}$ be a family of random variables satisfying some distribution with a parameter $\alpha$, $E(X_{\alpha})$ be the expectation, and $Var(X_{\alpha})$ be the variance. In this paper, we study the infimum values of three…

We provide a lower bound on the probability that a binomial random variable is exceeding its mean. Our proof employs estimates on the mean absolute deviation and the tail conditional expectation of binomial random variables.

We give an elementary proof of the fact that a binomial random variable $X$ with parameters $n$ and $0.29/n \le p < 1$ with probability at least $1/4$ strictly exceeds its expectation. We also show that for $1/n \le p < 1 - 1/n$, $X$…

Chv\'{a}tal's conjecture in extremal combinatorics asserts that for any decreasing family $\mathcal{F}$ of subsets of a finite set $S$, there is a largest intersecting subfamily of $\mathcal{F}$ consisting of all members of $\mathcal{F}$…

A starting point in the investigation of intersecting systems of subsets of a finite set is the elementary observation that the size of a family of pairwise intersecting subsets of a finite set [n]={1,...,n}, denoted by 2^{[n]}, is at most…

Let $n\geq 2$ and $(X_i,1\leq i\leq n)$ be a centered Gaussian random vector. The Gaussian minimum conjecture says that $E\left(\min_{1\leq i\leq n}|X_i|\right)\geq E\left(\min_{1\leq i\leq n}|Y_i|\right)$, where $Y_1,\ldots,Y_n$ are…

We give the proof of a tight lower bound on the probability that a binomial random variable exceeds its expected value. The inequality plays an important role in a variety of contexts, including the analysis of relative deviation bounds in…

We generalise the known fact that for binomial $X_{n,k} \sim \mathrm{Bin}(n, k/n)$ one has $\inf_{k>1,n} \mathrm{P}(X_{n,k} \geq k) \geq \lim_{k \to 1+}\mathrm{P}(X_{2,k} \geq k) = 1/4$ to cover probabilities of exceeding a constant shift…

A degree-$d$ polynomial $p$ in $n$ variables over a field $\F$ is {\em equidistributed} if it takes on each of its $|\F|$ values close to equally often, and {\em biased} otherwise. We say that $p$ has a {\em low rank} if it can be expressed…

Let $A$ be a $n \times n$ symmetric matrix with $(A_{i,j})_{i\leq j} $, independent and identically distributed according to a subgaussian distribution. We show that $$\mathbb{P}(\sigma_{\min}(A) \leq \varepsilon/\sqrt{n}) \leq C…

In this paper a simple proof of the Chebyshev's inequality for random vectors obtained by Chen (arXiv:0707.0805v2, 2011) is obtained. This inequality gives a lower bound for the percentage of the population of an arbitrary random vector X…

We study a new family of random variables, that each arise as the distribution of the maximum or minimum of a random number $N$ of i.i.d.~random variables $X_1,X_2,\ldots,X_N$, each distributed as a variable $X$ with support on $[0,1]$. The…