智慧即财富
陈立功的文学城博客：驰纵骋横，谈今博古，飞花扬月，行文交友
关于随机变量与概率论之间的一个关系
2012/08/02日记
随机变量是统计学和概率中最重要的概念。在整个数理统计学领域有一种说法是，概率论是统计学的基础，而测度论是概率论的基础，由此，统计学被称为了一门纯粹的数学分支学科。换句话说，这意味着一个非数学背景出生的人将没有可能性在统计学的方法论领域做出有实质意义的贡献。他们将不会被那些数学背景的统计学家们放在眼里。
概率论以纯数学的语言对随机变量作了一种数学意义上的抽象而又严格的定义和解释：一个随机变量是定义在其概率空间上的一个可测函数。这个概念的定义在非数学背景的统计学家们看来是一个无法被直观理解或晦涩的陈述。
其实，我们应该知道，一个随机变量并非存在于概率论中，而是存在于现实世界里，而现实世界是一个直观且容易被一般人类的智力所理解的存在。概率论不过是在基于某种关于现实世界中随机变量的基本认识的基础上给出的一种理论性的解释。一旦关于随机变量的基本认识得到深化和发展，概率论中关于它的理论性解释也就应该会被改变。因此，当一个人谈论关于随机变量是什么之类的问题时，他 / 她不应该直接从概率论中取用当前的定义，而是必须将自己的注意力聚焦于现实世界中的随机变量，因为一个随机变量并非来源于概率论，而是出自现实世界；而现实世界也并非是从数学理论体系中演绎出来的，而是恰恰相反。是的，数学不过是人类通过自己的智慧对现实世界的一种理论模拟，且其继承和秉持的“严谨性”原则常常会禁锢人类对外部现实世界的观察与思考的灵活性和颠覆性。此外，更为不幸的是，人类的智慧在认识现实世界时可能会常常犯错误，因此，作为一个理论系统，在犯错误的可能性方面，数学本身也不例外。
A Relationship between Random Variable and Theory of Probability
Random variable is the most important concept in Statistics and the Theory of Probability. In the domain of Mathematical Statistics, a statement is popular that the Theory of Probability is the foundation of Statistics, and the Measure Theory is the foundation of Theory of Probability, that is to say, Statistics is considered as a pure branch of Mathematics. In other words, this means that a non-mathematical-background statistician is certainly unable to make a really significant contribution in the field of statistical methodology. He / she will be looked down by those mathematical-background statisticians.
In a pure mathematical language, the Theory of Probability gives us a sort of rigorous definition and explanation on this abstract concept in a mathematical sense: A random variable is a measurable function defined over a probability space. However, this is an obscure statement that may not be understood intuitively by those non-mathematical-background statisticians.
In fact, we should understand that, a random variable does not exist in the Theory of Probability but in the real world, and the real world is intuitive and easily to be understood by an ordinary intelligence of the human being. The Theory of Probability just gives a kind of theoretical explanation to it based on a basic knowledge about random variables in the real world. Once the knowledge is deepened and developed, the theoretical explanation in the Theory of Probability should be changed, too. Therefore, when someone talks about "what a random variable is", he / she should not take the current definition from the Theory of Probability, but must focus on the random variable in a real world, because a random variable is not derived from the Theory of Probability but from the real world; and the real world is not deduced from the theoretical system of Mathematics but in reverse. Yes, Mathematics is just a theoretical simulation to the real world by the human being's intellihence, and the principle of "rigorousness" inherited and upholded by Mathematics often detained flexibility and subversiveness of human being's observation and thinking on the external real world. In addition, more unfortunately, the intelligence may often make mistakes when it realizes the real world in its own languages. Therefore, as a theoreitcal system, even Mathematics itself is not an exception in making mistakes.
You current translation for "finest" is accurate.
Let's look at the finite discrete case. Instead of giving rigorous definition, propositions and proofs, I will present an example which I think would give a more intuitive feel for the concept involved. Your question regarding the assignment of probability weight or measure would hopefully be resolved easier this way.
Let a 6 member finite set or sample space be represented as A = {1, 2, 3, 4, 5, 6}. We choose for A a finest partition, which is a set of disjoint non-empty subsets whose union is the origin set, P = {{1},{2,3},{4,5,6}}. The events space or the set of measurable set M is the set of all arbitrary unions of these previously defined component and partitioning subsets, e.g., {1,2,3} and {2,3,4,5,6}. We can assign a probability measure m this way, m(empty set) = 0, m({1}) = 0.5, m({2,3}) = 0.2, m({4,5,6}) = 0.3, all non-negative values so that the sum of them adds to 1. Also, m(u union v) = m(u)+m(v) for any disjoint u and v in the event space. The arithmetic operation of addition and subtraction of m defined on M is called a sigma algebra.
Having constructed a probability measure, we can define a measurable function f on A. A measurable function is a function whose domain of any set of value has to be measurable. In other words, f can be defined arbitrarily except f(x) = f(y) if x and y belongs to the same element in P. For example, it is necessary that f(4) = f(6) and f is not a measurable function if f(2) = -2.3 and f(3) = 5.4. This is what the word "finest" means. The finest partition specifies the highest resolution for distinguishing the subsets and for assigning function values. The function f thus defined is what we call a random variable.
Incidentally, if you give another partition, say P1 = {{1},{2,3},{4,5},{6}} which is finer than P, this sequence {P, P1} is called a filtration, which is used in defining stochastic processes.
再次对你的热心参与和认真的交流表示衷心的感激。借此机会我想要进一步说明的是，我的英文很差，我也没有在英语国家接受教育的经历。我之所以重写你的那段话，并非是要修正你的文法，而不过是为了要使之更符合我自己能够容易理解的英文形式。如果有何冒犯，请原谅。
>The above two statements are not exactly equivalent. It is the "different events" that are "corresponding to set union and set difference of the states" not "the probabilities of different event" as your rewrite would implicate.<
You are right. I made a mistake. But, I am still confused by your statements: 你的原文先是说"It also assigns probability weights to these components", 然后却说"so that we can perform arithmetic operations such as addition and subtraction on the probabilities of different events"。请问，你是如何将原本赋予给components的probability weights转移到"different events"上而成为后者的probabilities的？
那句话我想作出以下修改：“它还要对这些组成部分赋予概率权重，从而我们可以对所有状态中对应着同集和差集的不同事件的概率作一些诸如加减乘除等的数学运算。”
另外，我还打算将finest翻译为“最精细的”、“不能再分的”。你认为如何？
I apologize for choosing to write in English as it is more conducive to keyboard strokes for someone who is not versed in speed typing of Chinese characters.
"It also assigns the probability weights to these components so that we can perform arithmetic operations such as addition and subtraction on the probabilities of different events corresponding to set union and set difference of the states."
Your rewrite:
"It also assigns a probability weight or probability to each component, and the probabilities of different events are corresponding to set union and set difference of the states, so that we can perform arithmetic operations, such as addition and subtraction, on the probabilities."
The above two statements are not exactly equivalent. It is the "different events" that are "corresponding to set union and set difference of the states" not "the probabilities of different event" as your rewrite would implicate. I acknowledge though structure-wise without resorting to context, it is a bit confusing whether "corresponding …" is qualifying "the probabilities" or "events".
Your rewrite "It …, and the probabilities … are …, so that " does not constitute good syntax.
It is correct to say "the concept is much easier to grasp …" while "the concept is much easier to be grasped …" as you put it is rather awkward. To make it clear, my original statement is equivalent to "it is much easier to grasp the concept …" rather than "the concept is to be grasped …".
In my original comment, it is better to delete the article "the" in "It also assigns the probability weights…". This is what you get when you do things in haste. :-)
The confusion may well be caused by my lengthy sentence, which violates the usual admonition of technical writing. On the other hand, I was trying to get the gist of my contention across quickly, albeit in hand-waving manner without getting into much details, in my first comment testing the water, since I do not know what the reaction would be.
Thanks very much for your comments. It is very helpful to me. Let me try to rewrite the paragraphy to an equivalent one and then translate it into Chinese. 如果我的改写和翻译存在偏离原文之处，请原作者予以指正。
The mathematical definition aptly and rigorously delineates what we would like to capture with the concept of "random variable" in "reality". Measurability specifies the states one would like to consider. Most important, a measure specifies the finest and disjoint components of states upon which all other events build. It also assigns a probability weight or probability to each component, and the probabilities of different events are corresponding to set union and set difference of the states, so that we can perform arithmetic operations, such as addition and subtraction, on the probabilities. The concept is much easier to be grasped when looking at the discrete sample space. The case for the continuum is a bit harder without preliminary knowledge of mathematical analysis particularly measure theory. However, aside from technical machineries, the essential idea especially the motivation is no different from the discrete case.
数学上的定义恰当而严格地勾勒出了我们试图在现实世界中捕捉到的“随机变量”的概念。（一个随机变量的）的可测性所规定的各种状态是我们要考虑到的。最重要的是，一个测度规定了由所有其它事件构成的各种状态中最好的和不相交的组成成分。它还要对这些组成部分赋予概率权重，这些不同事件的概率分别对应着所有状态中的同集和差集，从而我们可以对它们作一些诸如加减乘除等的数学运算。当我们考察离散样本空间时，这一数学概念很容易被理解。但是，如果没有基本的数学分析特别是测度论方面的知识的话，对于连续空间的随机变异的理解则会显得比较困难。然而，除了技术因素，关于连续性随机变量的基本思想，尤其是关于如何认识它的动机，与离散的情形并没有什么差别。
我的评论：
原文在此使用了很多在其所涉及的范畴内没有严格定义的名词，诸如，state, event, component, set union, set difference, probability weight, probability, 等等，这容易引起误解和混淆。如果可能的话，希望原作者能一一解释它们之间的异同。
我不懂概率论，所以，我无法深刻地理解那个数学定义。但我确实懂一点统计，且对随机变量有一点理解力，因而我需要有自己的定义。
我问过Dr. Efron这个问题，他回答说，random variable does mean something is randomly variable. "Something" 在这里是一个名词，但我们显然不能用它来进行概念抽象和定义。我的建议是attribute。
如果attribute能够被接受，那么，关于它的抽象定义就应该解决以下几个基本问题：
What is an attribute?
Why is it variable?
How is it variable？ = Why its variable is randomly?
我不纠结。数学上如何定义随机变量是数学家的任务，但这个定义不应排斥其它领域的人给出一个抽象的定义。
我正好有一个问题，借此请教你一下：
如果用randomly (adverb) + variable (adjective) + noun (名词)取代random (adjective) + variable (noun)，请问，哪一个英文和中文名词最恰当？
如果可以用 randomly variable noun 取代 random variable，如何定义那个noun，请问这是一个数学定义，还是一个哲学定义？
这论那论很多都是不同实物的抽象，不抽象连1，2，3，4...概念都出不来。
“...一个人谈论1，2，3，4...之类的问题时，他/她不应该直接从算术中取用1，2，3，4的定义，应该想着一个苹果、两个苹果...”