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cdux
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It seems weird that such a relatively complex concept is simply given as a definition in most textbooks and then dismissed for further explanation other than using it intact or as a basis for further proofs.
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The quirk is that in this particular case the concept is very intuitive [edit: not the derivation or the result] and it stinks from miles that it could be proven, and not surprisingly I found a derivation of it from more basic concepts on the wikipedia article linked in here.Stephen Tashi said:In mathematics, definitions are not proven. A definition has the form: "[Undefined statement] will be defined to mean [Defined statement]". It's simply a matter of human culture whether a definition is widely accepted. You can't prove a definition is true. It has the same status as an assumption.
If we have two definitions, we can investigate whether the two things that are defined are equivalent. If we have a definition, we can investigate whether the thing that is defined actually exists in the mathematical sense, by showing an example of it or trying to prove no example can exist. If you have something along those lines in mind, you should explain your question.
cdux said:
cdux said:I start suspecting that what Kolmogorov did was to omit a linguistic definition that he had in mind which was then on derived by him to that simple equation because he thought of it not as generally applicable as he'd like, so he went on with simply providing the formula itself as a definition..
Stephen Tashi said:The wikipedia article does not describe Kolmorgorov's approach to probability theory. It merely uses the terminology "Komogorov definition". Your suspicions about his method are too simplistic. If you study continuous random variables and measure theory, you can understand the Kolmogorov approach.
cdux said:I start suspecting that what Kolmogorov did was to omit a linguistic definition that he had in mind which was then on derived by him to that simple equation because he thought of it not as generally applicable as he'd like, so he went on with simply providing the formula itself as a definition. ... he wasn't a 'words' man as much as a 'numbers' man to say it simply.
Yes, the Kolmogorov definition of the conditional probability can be proven using the axioms of probability and basic properties of conditional probability.
The Kolmogorov definition of the conditional probability is a mathematical formula for calculating the probability of an event A given that another event B has occurred. It is defined as P(A|B) = P(A ∩ B) / P(B), where P(A ∩ B) represents the probability of both events A and B occurring together.
The Kolmogorov definition of the conditional probability is important because it allows us to calculate the probability of an event taking into account additional information or conditions. This is useful in many real-world applications, such as risk assessment and decision making.
The Kolmogorov definition of the conditional probability is unique in that it is derived from the axioms of probability and does not rely on any assumptions or additional properties. Other definitions, such as the Bayes' theorem, may require additional assumptions or conditions.
One limitation of the Kolmogorov definition is that it assumes that the events A and B are independent, meaning that the occurrence of one event does not affect the probability of the other. In real-world scenarios, this may not always be the case and can lead to inaccurate calculations of conditional probability.