智慧即財富
陳立功的文學城博客：馳縱騁橫，談今博古，飛花揚月，行文交友
關於隨機變量與概率論之間的一個關係
2012/08/02日記
隨機變量是統計學和概率中最重要的概念。在整個數理統計學領域有一種說法是，概率論是統計學的基礎，而測度論是概率論的基礎，由此，統計學被稱為了一門純粹的數學分支學科。換句話說，這意味著一個非數學背景出生的人將沒有可能性在統計學的方法論領域做出有實質意義的貢獻。他們將不會被那些數學背景的統計學家們放在眼裏。
概率論以純數學的語言對隨機變量作了一種數學意義上的抽象而又嚴格的定義和解釋：一個隨機變量是定義在其概率空間上的一個可測函數。這個概念的定義在非數學背景的統計學家們看來是一個無法被直觀理解或晦澀的陳述。
其實，我們應該知道，一個隨機變量並非存在於概率論中，而是存在於現實世界裏，而現實世界是一個直觀且容易被一般人類的智力所理解的存在。概率論不過是在基於某種關於現實世界中隨機變量的基本認識的基礎上給出的一種理論性的解釋。一旦關於隨機變量的基本認識得到深化和發展，概率論中關於它的理論性解釋也就應該會被改變。因此，當一個人談論關於隨機變量是什麽之類的問題時，他 / 她不應該直接從概率論中取用當前的定義，而是必須將自己的注意力聚焦於現實世界中的隨機變量，因為一個隨機變量並非來源於概率論，而是出自現實世界；而現實世界也並非是從數學理論體係中演繹出來的，而是恰恰相反。是的，數學不過是人類通過自己的智慧對現實世界的一種理論模擬，且其繼承和秉持的“嚴謹性”原則常常會禁錮人類對外部現實世界的觀察與思考的靈活性和顛覆性。此外，更為不幸的是，人類的智慧在認識現實世界時可能會常常犯錯誤，因此，作為一個理論係統，在犯錯誤的可能性方麵，數學本身也不例外。
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.
再次對你的熱心參與和認真的交流表示衷心的感激。借此機會我想要進一步說明的是，我的英文很差，我也沒有在英語國家接受教育的經曆。我之所以重寫你的那段話，並非是要修正你的文法，而不過是為了要使之更符合我自己能夠容易理解的英文形式。如果有何冒犯，請原諒。
>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: 你的原文先是說"It also assigns probability weights to these components", 然後卻說"so that we can perform arithmetic operations such as addition and subtraction on the probabilities of different events"。請問，你是如何將原本賦予給components的probability weights轉移到"different events"上而成為後者的probabilities的？
那句話我想作出以下修改：“它還要對這些組成部分賦予概率權重，從而我們可以對所有狀態中對應著同集和差集的不同事件的概率作一些諸如加減乘除等的數學運算。”
另外，我還打算將finest翻譯為“最精細的”、“不能再分的”。你認為如何？
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. 如果我的改寫和翻譯存在偏離原文之處，請原作者予以指正。
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.
數學上的定義恰當而嚴格地勾勒出了我們試圖在現實世界中捕捉到的“隨機變量”的概念。（一個隨機變量的）的可測性所規定的各種狀態是我們要考慮到的。最重要的是，一個測度規定了由所有其它事件構成的各種狀態中最好的和不相交的組成成分。它還要對這些組成部分賦予概率權重，這些不同事件的概率分別對應著所有狀態中的同集和差集，從而我們可以對它們作一些諸如加減乘除等的數學運算。當我們考察離散樣本空間時，這一數學概念很容易被理解。但是，如果沒有基本的數學分析特別是測度論方麵的知識的話，對於連續空間的隨機變異的理解則會顯得比較困難。然而，除了技術因素，關於連續性隨機變量的基本思想，尤其是關於如何認識它的動機，與離散的情形並沒有什麽差別。
我的評論：
原文在此使用了很多在其所涉及的範疇內沒有嚴格定義的名詞，諸如，state, event, component, set union, set difference, probability weight, probability, 等等，這容易引起誤解和混淆。如果可能的話，希望原作者能一一解釋它們之間的異同。
我不懂概率論，所以，我無法深刻地理解那個數學定義。但我確實懂一點統計，且對隨機變量有一點理解力，因而我需要有自己的定義。
我問過Dr. Efron這個問題，他回答說，random variable does mean something is randomly variable. "Something" 在這裏是一個名詞，但我們顯然不能用它來進行概念抽象和定義。我的建議是attribute。
如果attribute能夠被接受，那麽，關於它的抽象定義就應該解決以下幾個基本問題：
What is an attribute?
Why is it variable?
How is it variable？ = Why its variable is randomly?
我不糾結。數學上如何定義隨機變量是數學家的任務，但這個定義不應排斥其它領域的人給出一個抽象的定義。
我正好有一個問題，借此請教你一下：
如果用randomly (adverb) + variable (adjective) + noun (名詞)取代random (adjective) + variable (noun)，請問，哪一個英文和中文名詞最恰當？
如果可以用 randomly variable noun 取代 random variable，如何定義那個noun，請問這是一個數學定義，還是一個哲學定義？
這論那論很多都是不同實物的抽象，不抽象連1，2，3，4...概念都出不來。
“...一個人談論1，2，3，4...之類的問題時，他/她不應該直接從算術中取用1，2，3，4的定義，應該想著一個蘋果、兩個蘋果...”