概率论是研究随机性或不确定性等现象的 数学

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概率论起源于17世纪的法国,当时两位伟大的法国数学家,布莱斯·帕斯卡和皮埃尔·德·费马,对两个来自机会博弈的问题进行了通信。帕斯卡和费马解决的问题继续影响着惠更斯、伯努利和DeMoivre等早期研究者建立数学概率论。今天,概率论是一个建立良好的数学分支,应用于从音乐到物理的学术活动的每一个领域,也应用于日常经验,从天气预报到预测新的医疗方法的风险。

本文是为数学、物理和社会科学、工程和计算机科学的二、三、四年级学生开设的概率论入门课程而设计的。它提出了一个彻底的处理概率的想法和技术为一个牢固的理解的主题必要。文本可以用于各种课程长度、水平和重点领域。

在标准的一学期课程中,离散概率和连续概率都包括在内,学生必须先修两个学期的微积分,包括多重积分的介绍。第11章包含了关于马尔可夫链的材料,为了涵盖这一章,一些矩阵理论的知识是必要的。

文本也可以用于离散概率课程。材料被组织在这样一种方式,离散和连续的概率讨论是在一个独立的,但平行的方式,呈现。这种组织驱散了对概率过于严格或正式的观点,并提供了一些强大的教学价值,因为离散的讨论有时可以激发更抽象的连续的概率讨论。在离散概率课程中,学生应该先修一学期的微积分。

为了充分利用文中的计算材料和例子,假设或必要的计算背景很少。所有在文本中使用的程序都是用TrueBASIC、Maple和Mathematica语言编写的。

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Understanding the likelihood for an election to be tied is a classical topic in many disciplines including social choice, game theory, political science, and public choice. The problem is important not only as a fundamental problem in probability theory and statistics, but also because of its critical roles in many other important issues such as indecisiveness of voting, strategic voting, privacy of voting, voting power, voter turn out, etc. Despite a large body of literature and the common belief that ties are rare, little is known about how rare ties are in large elections except for a few simple positional scoring rules under the i.i.d. uniform distribution over the votes, known as the Impartial Culture (IC) in social choice. In particular, little progress was made after Marchant [Mar01] explicitly posed the likelihood of k-way ties under IC as an open question in 2001. We give an asymptotic answer to the open question for a wide range of commonly-studied voting rules under a model that is much more general and realistic than i.i.d. models including IC--the smoothed social choice framework [Xia20], which was inspired by the celebrated smoothed complexity analysis [ST09]. We prove dichotomy theorems on the smoothed likelihood of ties under a large class of voting rules. Our main technical tool is an improved dichotomous characterization on the smoothed likelihood for a Poisson multinomial variable to be in a polyhedron, which is proved by exploring the interplay between the V-representation and the matrix representation of polyhedra and might be of independent interest.

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