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.
nightrider 发表评论于
TNEGI//ETNI, no response?
nightrider 发表评论于
You are most welcome. I apologize if I gave the impression that I was piqued by your rewriting of my comment. There is no need to apologize as no offense was taken. I only wanted to clarify more precisely what I intended to say.
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.
>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？
Rather than carefully define all the terms used earlier as you rightly requested, I will later give a simple example in discrete sample space which I think will illustrate the concepts more intuitively than rigorous definitions. One can surely look up the definition and construction of probability measure in any decent textbook on probability and stochastic processes.
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.
nightrider 发表评论于
You mistranslated "the finest" as the best. Rather, here "fine" refers to resolution, particularly being minute, thin, not coarse. Specifically, "finest state" refers to the smallest atomic state which the observer can not or care not to divide any further.
nightrider 发表评论于
My original statement:
"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.
TNEGI//ETNI 发表评论于
回复nightrider的评论:
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.
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。
nightrider 发表评论于
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 the probability weights to these components so that we can perform arithmetic perations such as addition and subtraction on the probabilities of different events corresponding to set union and set difference of the states. The concept is much easier to grasp 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.
