概率解题中样本空间的选择

无论是古典概型中随机事件概率的计算,还是随机变量分布的确定,选择好适当的样本空间都是解题的关键.借助“草绳结环”和“圆周上均匀分布的期望和方差”两个经典的概率问题,说明巧妙地选取样本空间有利于概率问题的顺利解决.     

古典概率模型中样本空间的选择

古典概型是高中数学概率学习的核心．在古典概率的计算过程中，样本空间的选择是关键一环，主要表现在“能否正确选择样本空间”和“能否选择较小样本空间”两方面．在理解题意的过程中是否注重这两方面的思考，将决定解题的成功与高效．本文举例说明，供读者参考。     

概率论中审视样本空间的思维策略

构造恰当的样本空间是古典概率解题的第一步，选择样本空间的计量方法是解决问题的关键，对于同一个随机试验，样本空间可以相当简单，也可以相当复杂，下面通过实例，谈谈在保证等可能性的前提下，通过改变样本容量计量方法或缩减样本空间使得概率计算化繁为简的思维策略。     

浅谈条件概率的计算问题

概率论中的条件概率是这样定义的，设A，B是两个事件，且P（A）＞0，则称P（B|A）＝P（AB）／P（A）为事件A发生条件下事件B发生的条件概率。下面列出计算条件概率P（BA）的三种方法，并举例进行讨论和说明。1．在样本空间D中，先计算P（AB），P（A），再按照定义计算；2．在样本空间o的缩减样本空间见中计算B发生概率，即P（B／A），这里，D。二QuA3．按贝叶斯公式计算。例1将一枚硬币抛掷三次，记事件A为“至少出现一个正面”，记事件B为“至少出现两个反面”，求P（B／A）与P（AB）。阐显然，AB表示“恰有一个正面二个反…     

如何正确理解两事件的相互独立性

通过实例说明两事件的相互独立性与样本空间Ω的结构有关,与概率测度P的定义方式有关。     

概率论教学札记

<正> 一关于事件和基本事件样本空间的一个子集是否为事件,依赖于事件域的选取,由样本点构成的单点子集也可以不是事件。我们认为,基本事件这个术语以不用为宜。学生常常望文生义地认为基本事件不但一定是事件,而且是基本事件。尽管以后反复说明“基本事件也可以不是事件”,但先入之     

统计流形上的某些不变性

§1.引论 Efron(1975)曾经证明,他所定义的统计曲率在参数空间中参数的单调变换下以及样本空间中充分统计量的映射下保持不变.这些不变性说明,Efron曲率是分布族固有的性质,与参数的选择无关;而且我们可以选取一个充分统计量代替原有样本,讨论与统计曲率有关的问题,这正是研究统计问题时所希望的.Amari(1982)和Bates and Watts(1980)分别定义了另外两种统计曲率.他们对上述不变性问题未曾予以讨论.本文首先结合Amari法曲率     

概率论札记

<正> 样本空间的一个子集是否为事件,依赖于事件域的选取,由样本点构成的单点子集也可以不是事件.我们认为,基本事件这个术语以不用为宜.学生常常望文生义地认为基本事件不但一定是事件,而且是基本事件.尽管以后反复说明“基本事件可以不是事件”.但先入之见已很难扭转过来了.这同述语本身语义上的混乱有关.[1],[2]未采用基本事件这一术语,效果     

概率论中审视样本空间的思维策略

构造恰当的样本空间是古典概率解题的第一步,选择样本空间的计量方法是解决问题的关键.对于同一个随机试验,样本空间可以相当简单,也可以相当复杂.下面通过实例,谈谈在保证等可能性的前提下,通过改变样本容量计量方法或缩减样本空间使得概率计算化繁为简的思维策略.1改变样本空间的计量方法,实现无限向有限的转化在古典概型中,试验的结果是有限的,当试验结果为无限时会出现本质上的困难.此时,我们可用某种数量特征(如长度,面积)来表示总和,设为S;并且其中的一部分,即有利于随机事件A的基本事件数,也可以用同样的数量特征来表示,设为s;则随…     

恰当选取样本空间，简化古典概率计算

用概率的古典定义计算概率时 ,首先要确定随机试验是什么 ,从而确定出样本空间 .若样本空间中的各基本事件在试验中的出现是等可能的 ,则可由古典概率公式求各随机事件的概率 .但同一问题随试验的内容不同可选取不同的样本空间 ,只要满足样本空间中的基本事件只有有限个 ,且它们的出现是等可能的 ,就可用古典概率公式计算 ,且计算出的结果必定相同 .因此试验的样本空间选得好 ,问题解决起来就会简便一点 .下面举例说明 .在下面的例子中均以 N表示基本事件总数 ,M表示所求事件包含的基本事件数 .例 1　袋内有 a个白球与 b个黑球 ,每次从袋中…     

Single Sample Path-Based Optimization of Markov Chains

Motivated by the needs of on-line optimization of real-world engineering systems, we studied single sample path-based algorithms for Markov decision problems (MDP). The sample path used in the algorithms can be obtained by observing the operation of a real system. We give a simple example to explain the advantages of the sample path-based approach over the traditional computation-based approach: matrix inversion is not required; some transition probabilities do not have to be known; it may save storage space; and it gives the flexibility of iterating the actions for a subset of the state space in each iteration. The effect of the estimation errors and the convergence property of the sample path-based approach are studied. Finally, we propose a fast algorithm, which updates the policy whenever the system reaches a particular set of states and prove that the algorithm converges to the true optimal policy with probability one under some conditions. The sample path-based approach may have important applications to the design and management of engineering systems, such as high speed communication networks.This work was supported in part by     

Implementation of sensitivity calculations on a monte carlo experiment

A new approach to the optimization of discrete-event dynamic systems has recently been developed (Refs. 1–4). The implementation of this approach requires answering the following question: given an observed value of the outcome of a random variable, what would have been the outcome if the parameters of the random variable had been different? The answer to this question would traditionally involve the value of an outcome in an underlying sample space. However, this underlying value cannot normally be observed. In this note, we give a framework for answering this question, in terms of the observed value alone. This point had not been considered rigorously in the new approach of Refs. 1–4, and our note derives a basic equation required for that approach.     

Exact point-distributions over the complex sphere

We study point-distributions over the surface of the unit sphere in unitary space that generate quadrature rules which are exact for spherical polynomials up to a certain bi-degree. In this first stage, we explore several different characterizations for this type of point sets using standard tools such as, positive definiteness, reproducing kernel techniques, linearization formulas, etc. We find bounds on the cardinality of a point-distribution, without discussing the deeper question regarding best bounds. We include examples, construction methods and explain, via isometric embeddings from real to complex spheres, the proper connections with the so-called spherical designs.     

Principal Normal Indicatrices of Closed Space Curves

A theorem due to J. Weiner, which is also proven by B. Solomon, implies that a principal normal indicatrix of a closed space curve with nonvanishing curvature has integrated geodesic curvature zero and contains no subarc with integrated geodesic curvature . We prove that the inverse problem always has solutions if one allows zero and negative curvature of space curves and explain why this not is true if nonvanishing curvature is required. This answers affirmatively an open question asked by W. Fenchel in 1950 under the above assumptions but in general this question is found to be answered to the negative.     

Extended dynamic partial-overlapping batch means estimators for steady-state simulations

Estimating the variance of the sample mean from a stochastic process is essential in assessing the quality of using the sample mean to estimate the population mean, which is the fundamental question in simulation experiments. Most existing studies for estimating the variance of the sample mean from simulation output assume that the simulation run length is known in advance. An interesting and open question is how to estimate the variance of the sample mean with limited memory space, reasonable computation time, and good statistical properties such as small mean-squared-error (mse), without knowing the simulation run length a priori. This paper proposes a finite-memory algorithm that satisfies the above good estimation criteria. Our findings show that the proposed algorithm improves over its competitors in terms of the mse criterion.     

Monotone retractability and retractional skeletons

We prove that a countably compact space is monotonically retractable if and only if it has a full retractional skeleton. In particular, a compact space is monotonically retractable if and only if it is Corson. This gives an answer to a question of R. Rojas-Hernández and V.V. Tkachuk. Further, we apply this result to characterize retractional skeleton using a topology on the space of continuous functions, answering thus a question of the first author and a related question of W. Kubi?.     

A Gâteaux differentiability space that is not weak Asplund

In this paper we construct a Gâteaux differentiability space that is not a weak Asplund space. Thus we answer a question raised by David Larman and Robert Phelps from 1979.

     

Henstock-Kurzweil可积函数空间的紧性特征

本文研究Henstock-Kurzweil可积(HK可积)函数空间中的一个经典问题.文章通过研究分布Henstock-Kurzweil积分(DHK积分)的性质,给出了该问题的否定答案.进一步,利用收敛性获得了函数HK可积的一个充分必要条件.最后,在上述结论的基础上刻画了HK可积函数空间的紧性.所得结果丰富和推广了HK可积函数空间理论.     

The rational line partitions every self-dense metrisable space

In the present note we shall prove that a metrisable space can be partitioned by the rational line if that space is self-dense. This gives an affirmative answer to a question raised by Rankston-McGovern [1].     

强零维度量空间的刻画

本文研究了强零维度量空间,借助点星弱邻域网刻画了强零维度量空间,获得了强零维度量空间等价于具有离散覆盖列的点星弱邻域网的非空正则空间,回答了葛英提出的一个问题.