1.7: Summary
- Page ID
- 44603
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This chapter started with an introduction into the correspondence between probability theory and real-world experiments involving randomness. While almost all work in probability theory works with established probability models, it is important to think through what these probabilities mean in the real world, and elementary subjects rarely address these questions seriously.
The next section discussed the axioms of probability theory, along with some insights about why these particular axioms were chosen. This was followed by a review of conditional probabilities, statistical independence, random variables, stochastic processes, and expectations. The emphasis was on understanding the underlying structure of the field rather than reviewing details and problem solving techniques.
This was followed by a fairly extensive treatment of the laws of large numbers. This involved a fair amount of abstraction, combined with mathematical analysis. The central idea is that the sample average of n IID rv’s approaches the mean with increasing n. As a special case, the relative frequency of an event A approaches Pr{A}. What the word approaches means here is both tricky and vital in understanding probability theory. The strong law of large numbers and convergence WP1 requires mathematical maturity, and is postponed to Chapter 4 where it is first used.
The final section came back to the fundamental problem of understanding the relation between probability theory and randomness in the real-world. It was shown, via the laws of large numbers, that probabilities become essentially observable via relative frequencies calculated over repeated experiments.
There are too many texts on elementary probability to mention here, and most of them serve to give added understanding and background to the material in this chapter. We recommend Bertsekas and Tsitsiklis [2], both for a careful statement of the fundamentals and for a wealth of well-chosen and carefully explained examples.
Texts that cover similar material to that here are [16] and [11]. Kolmogorov [14] is readable for the mathematically mature and is also of historical interest as the translation of the 1933 book that first put probability on a firm mathematical basis. Feller [7] is the classic extended and elegant treatment of elementary material from a mature point of view. Rudin [17] is an excellent text on measure theory for those with advanced mathematical preparation.