关于随机变量与概率论之间的一个关系
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.