| 摘要: |
| 设$B(n, p)$表示服从参数为$n$和$p$的二项分布的随机变量. Chv\'{a}tal 定理的含义为：对任意一个固定的$n\ge 2,$ $m$的取值范围为${0,\ldots, n},$ 当$m$ 最接近于$\frac{2n}{3}$ 时, 概率$q_m:=P(B(n,m/n)\le m)$取到最小值. 受此定理的启发, 在本文中我们考虑概率$P(X\le \kappa E[X])$的下确界, 其中$\kappa$是一个正实数, 随机变量$X$ 的分布属于无穷可分分布, 包括逆高斯分布, 对数正态分布, Gumble 分布和Logistic 分布. |
| 关键词: Chv\'{a}tal 定理 无穷可分分布 逆高斯分布 对数正态分布 Gumbel 分布 Logistic 分布 |
| DOI： |
| 分类号:0211 |
| 基金项目: |
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| The infimum values of the probability functions for some infinitely divisible distributions motivated by Chv\'{a}tal's theorem |
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Hu Zechun1, Zhou Qianqian2, 周兴旺
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1.Sichuan University;2.Nanjing University of Posts and Telecommunications
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| Abstract: |
| 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 closest to $\frac{2n}{3}$. Motivated by this theorem, in this paper we consider the infimum value of the probability $P(X\leq \kappa E[X])$, where $\kappa$ is a positive real number, and $X$ is a random variable whose distribution belongs to some infinitely divisible distributions including the inverse Gaussian, log-normal, Gumbel and logistic distributions. |
| Key words: Chv\'{a}tal's theorem Infinitely divisible distribution Inverse Gaussian distribution Log-normal distribution Gumbel distribution Logistic distribution. |
