What's the value of a random variable?
Do you mean: "What are random variables good for" ?
Or do you mean the "value" in the sense [itex]X(\omega) [/itex] where [itex] X[/itex] is a random variable and [itex] \omega[/itex] is a sample point?
[itex]X(\omega) [/itex] doesn't really seem like a mathematical function but the result of an experimental trial.
Probability theory was given an axiomatic setting by Kolmogoroff in the 1930's. It looks in some ways like measure theory.
The experimental trial can be conceptual, like analysis of what happens after a series of coin flips by an ideal coin.
Thanks mathman. I've since been reading about axiomatic probability. My interest is in quantum mechanics where probabilities seem to have a central role.
So I am curious as to the answer to the question: Can probability theory be separated from physics (in this case classical stochastic processes), so that axiomatic quantum mechanics is, in part, defined upon classical uncertainty?
I don't know what you mean by separated.
This is the pivotal question, isn't it? I hadn't been sure at the time. Now, however:-
If the axioms and theorems of probability theory are so abstract as to be free of any reference to any physical system, then probability theory is separate from physics. If there is an explicit reliance they are not.
Does this sound meaningful to you?
There is the many universes interpretation. The sample space [itex]\Omega[/itex] has one point [itex]\omega[/itex] for each possible universe. When we do the experiment whose outcome is the random variable [itex]X[/itex], then [itex]X(\omega)[/itex] is the "result of an experimental trial" as you say. But it is different depending on which universe we are in. Thus we consider [itex]X[/itex] to be a function defined on [itex]\Omega[/itex].
Probability theory is a branch of mathematics. Physicists use it just like they use many other branches such as calculus, vector analysis, differential equations, etc.
Other fields where probability is used include statistics, finance, biology, etc.
That's not really what I meant. I'm struggling to express this question in an understandable way. This may be it: A computer algorithm may be one way to test independence of a mathematical system from physical props.
I may be mistaken, but can all axiomatic mathematical systems be encoded as computer algorithms? This should depend on what 'encoded' means. Arithmetic might be a good example to look at.
I should be able to take the axioms of arithmetic and compute new statements that are consistent with the axioms through combination.
Not so obvious is that any result such as 2+2=4 is a new statement when expressed as
[tex]\forall A,B \in R: ( A=2 \ \wedge \ B=2 \ \wedge \ C=A+B ) \rightarrow C=4[/tex]
I'm not sure the axioms of probability theory can be encoded by computer algorithm and obtain new and constant statements such as the addition of specific values in addition. We can’t use pseudo random number generators. That would be cheating.
Or maybe I’m looking at this wrong. Any thoughts?
g_edgar, you gave me a thought. Maybe the elements that can form new statements are sample spaces but not the points of the sample spaces.
Mathematics is independent of physical properties. Whether or not the axioms of a mathematical system can be encoded as computer algorithms is besides the point.
I guess you don't understand ordinary American jargon. Besides the point is another way of saying irrelevant.
In any case the axioms for probability theory are essentially the same as those for measure theory, with the additional condition that the total measure (probability) is one.
It is not irrelevent. If an axiomatic system cannot be encoded in a symbolic language, it is flawed.
I have never tried to and I don't care to. I'll presume it is doable.
Since the Kolmogoroff axioms as well as measure theory have been in use for a very long time, I strongly believe that they are not flawed.
By encoding, do you mean the automation of the logic of probability theory (PT) to prove theorems? If so, the answer is yes. There is no flaw in PT if you accept the axioms and definitions. The definition of a random variable in particular, is that it is simply a mapping from a probability space to an event space with no guidance at all as to what value the function assigns to an event provided it is in the interval [0,1]. The following link is book sized, but the preface should at least shed some light on your question. The basic thrust is that the idea of "randomness" should be replaced by "incomplete information". There is also a good article (IMHO) on Quantum Logic and Probability Theory in the Stanford Encyclopedia of Philosophy.
I'm sorry I can't reply intelligently to your post. I'll have to satisfy for myself that PT is consistent, unflawed or otherwise by putting in the effort.
As I stated earlier, my question is indirectly about probability in (formalized) quantum mechanics. So in addition, consistency between the two is also a subject of interest.
My use of the word 'encoded' was trite. Symbolically encoded, by rules of substitution, two true statements should yield another true statement under the set of axioms.
I've downloaded your recommended article.
That's an unrealistic goal. No one has proved the axioms of ZFC are self consistent or "unflawed".
Thanks SW. I'll keep ZFC in mind. My quest in this is not a large priority--just something that has nagged me on occasion over 10 to 15 years or so.
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