1 问题由来 看到近来群里在讨论如下的概率问题, 例1 有M种卡片,每种卡片数量无限,每次随机从M种卡片中选择一张,选择到每种卡片的概率相同.现在需要集齐M种卡片,即每种卡片至少需要一张,集齐之后则终止.问:选择n次(n≥M),能够集齐M张卡片的概率是多少? 这个问题提得挺有趣,设问也看似合理,但是,要解答它却非易事.这是一类等待胜利问题,而且是带有参数变化的等待问题,已经远远超出中学数学中关于概率论的知识范围.

2018年高校招生全国统一考试理科数学试卷Ⅰ的第20题是一道关于产品检验的题目.题目开宗明义,面对的是大批量的产品.产品成箱包装,每箱200件.每箱先抽检20件,根据检验结果决定是否对余下的所有产品作检验.命题者怕在这里引起误会,不可谓不仔细地斟酌了词句,特别强调了要决定的是“是否对余下的所有产品作检验”,说明这里的选项只有两个:要么对余下的所有产品都作检验(全检),要么就都不检验(全不检),不存在部分检验的问题.正是在这种思路的指导之下,诞生了对于第(2)小题(ii)的官方解答.

1.某甲沿着环状道路以常速跑步.在该环状道路上安装着两个摄像头.在他起跑两分钟时接近了第一个摄像头.试问:他跑一圈需要多少时间? 2.点E在平行四边形ABCD内部,使得CD=CE.证明:直线DE垂直于经过线段AE与线段BC中点的直线.

1.是否存在这样的正整数n,使n,n2,n3的首位数字彼此相同,并且不是1 ? 2.沿着圆周按某顺序摆放着正整数1到1 000,使得其中每个数都是自己的两个邻数的和的约数.现知,正整数k的两侧邻数都是奇数,试问:k的奇偶性可能如何? 3.厨师每天烘制一个尺寸为3×3的正方形蛋糕.甲立即从上面为自己切下4块尺寸为1×1的部分,它们的边都平行于蛋糕的边,但未必沿着3×3方格网的网线.此后,乙从剩下的蛋糕上为自己切下一块尽可能大的正方形部分,它的边也平行于蛋糕的边.

1.是否存在这样的正数n,使n,n2,n3的首位数字彼此相同,并且不是1? 2.沿着圆周按某顺序摆放着正整数1到1 000,使得其中每个数都是自己的两个邻数的和的约数.现知,正整数k的两侧邻数都是奇数,试问:k的奇偶性可能如何?

(八年级) 1.维佳试图找出一个算式,其中有数1,有括号,有加号和乘号,使得: (1)算式的值等于10; (2)如果将算式中的所有加号都换成乘号,而所有的乘号都换成加号,算式的值仍然是10. 试给出这样的算式的一个例子. 2.将一条折线称为蛇状折线,如果它的所有相邻节之间的夹角全都彼此相等,并且除了头尾两节之外,其余各节的两个相邻节都分别位于该节的两个不同侧(即位于由经过该节的直线所分成的两个不同的半平面中).蛇状折线的一个例子如图1所示.

We investigate the precise large deviations of random sums of negatively dependent random variables with consistently varying tails. We find out the asymptotic behavior of precise large deviations of random sums is insensitive to the negative dependence. We also consider the generalized dependent compound renewal risk model with consistent variation, which including premium process and claim process, and obtain the asymptotic behavior of the tail probabilities of the claim surplus process.

Several limit laws for the Zagreb indices of the classical Erdös–Rényi random graphs are investigated in this paper. We have obtained the necessary and sufficient condition for the asymptotic normality of the two Zagreb indices (suitably normalized), as well as the explicit values for the means and variances of both the indices. Besides, the limiting joint distribution of the numbers of paths of various lengths is also studied under several conditions.

Scale-free graphs have many applications in complex networks,such as the "WWW" that has led to the emerging role of random networks.The degree sequences of Buckley-Osthus model were studied.Firstly,the asymptotic property for the vertices with indegree d was obtained,then the convergence of the maximal degree of vertices as the size of graph grows to infinity was investigated.

概率统计内容进入中小学数学课堂是近几年的新生事物,是与国际接轨的一项举措,符合与时俱进的精神. 但是,综观整个中小学阶段的教学内容安排,却令人感觉杂乱无章,缺乏统一考虑和整体安排.有些内容反复出观,有些内容缺乏深度.

This paper deals with the approximation of the tail probability of randomly weighted sums of a sequence of pairwise quasi-asymptotically independent but non-identically distributed dominatedly-varying-tailed random variables. The weights are independent of the former sequence, satisfying some assumptions about the moments. But no requirements on the dependence structure of the weights are imposed.

The domination relationship between non-negative distributions is an important question in applied probability. It has important applications in the fields of finance, insurance and risk theory. In this paper, based on class ℳ, we find the sufficient condition of dominating all light-tailed distributions and also discuss its necessity. Almost all heavy-tailed distributions often used in risk theory satisfy this condition. We also consider the domination problem between heavy-tailed distributions, and show that classes and ℳ * have many good properties on domination problems.

The one-sided interval trees can be generated by division a given interval in various ways. The corresponding interval trees have difference properties. The maximal gap in the corresponding interval was considered. Some functional equations for the asymptotic distributions of the length of the maximal gaps in difference ways of division were obtained. Using recurrence, the expressions of asymptotic distributions for several ways of division in some small intervals were also obtained.

主要研究大小为n的随机二叉搜索树上3种不同类型的顶点数目.分别以Xn,Yn和Zn表示树中含有0,1,2个子点的顶点的数目.在建立Xn递归关系式的基础上,得到了Xn的期望、方差和大数律,并用压缩法证得了Xn的渐近正态性.对于Yn和Zn,也得到了类似的结论.

Let F(x) be a distribution function supported on [0, ∞) with an equilibrium distribution function F e (x). In this paper we pay special attention to the hazard rate function r e (x) of F e (x), which is also called the equilibrium hazard rate (E.H.R.) of F(x). By the asymptotic behavior of r e (x) we give a criterion to identify F(x) to be heavy-tailed or light-tailed. Moreover, we introduce two subclasses of heavy-tailed distributions, i.e., ℳ and ℳ*, where ℳ contains almost all the most important heavy-tailed distributions in the literature. Some further discussions on the closure properties of ℳ and ℳ* under convolution are given, showing that both of them are ideal heavy-tailed subclasses. In the paper we also study the model of independent difference ξ = Z − θ, where Z and θ are two independent and non-negative random variables. We give intimate relationships of the tail distributions of ξ and Z, as well as relationships of tails of their corresponding equilibrium distributions. As applications, we apply the properties of class ℳ to risk theory. In the final, some miscellaneous problems and examples are laid, showing the complexity of characterizations on heavy-tailed distributions by means of r e (x).

This paper obtains the Embrechts-Veraverbeke asymptotic formula for the random walk with dependent steps, where the steps constitute a sequence of negatively associated random variables with a common heavy-tailed distribution such that its left tail is lighter than its right tail.

On the basis of an extension for an early result of Khintchine, the class of asymptotic distributions of the standardized Ψ-sums for a class of distributions is obtained in this paper.

第四轮 8.1.中午12点时,"西部仔"与"莫斯科仔"开始从相距90 km的两个地点分别开汽车以常速相向而行.2 h以后,他们的距离又是90 km.一个"不知情者"断言:西部仔在与莫斯科仔碰面之前以及莫斯科仔在与西部仔碰面之后,一共行驶了60 km.试证明:该断言不正确.

The length of the random path generated by the Bernoulli splitting algorithm was studied by means of Poisson transformation.For the first time the exact expression of this expectation was obtained.Based on this expression,the asymptotic analysis of the expectation of the random path were presented.Thus it was proved that as the set size n goes to infinite,the expectation of the random path length has the order of log2n.