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.